Timeline of category theory and related mathematics

This is a timeline of category theory and related mathematics. Its scope ('related mathematics') is taken as:

Categories of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

ic structures including representation theory
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

and universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....

;
Homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

;
Homotopical algebra
Homotopical algebra
In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases...

;
Topology using categories, including algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, categorical topology, quantum topology
Quantum topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of...

, low dimensional topology;
Categorical logic
Categorical logic
Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor...

and set theory in the categorical context such as algebraic set theory;
Foundations of mathematics building on categories, for instance topos theory;
Abstract geometry, including algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, categorical noncommutative geometry, etc.
Quantization related to category theory, in particular categorical quantization;
Categorical physics relevant for mathematics.

In this article and in category theory in general ∞ = ω.

Timeline to 1945: before the definitions

Year	Contributors	Event
1890	David Hilbert David Hilbert David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...	\| Resolution Resolution (algebra) In mathematics, particularly in abstract algebra and homological algebra, a resolution is a sequence of objects which is used to describe the structure of a module, or, more generally, the structure of an object in an abelian category.Generally, if the objects involved in the sequence have a... of modules and free resolution Resolution (algebra) In mathematics, particularly in abstract algebra and homological algebra, a resolution is a sequence of objects which is used to describe the structure of a module, or, more generally, the structure of an object in an abelian category.Generally, if the objects involved in the sequence have a... of modules.
1890	David Hilbert David Hilbert David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...	\| Hilbert's syzygy theorem Hilbert's syzygy theorem In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it... is a prototype for a concept of dimension in homological algebra Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and... .
1893	David Hilbert David Hilbert David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...	\| A fundamental theorem in algebraic geometry Algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex... , the Hilbert Nullstellensatz. It was later reformulated to: the category of affine varieties over a field k is equivalent to the dual of the category of reduced finitely generated (commutative) k-algebras.
1894	Henri Poincaré Henri Poincaré Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...	\| Fundamental group Fundamental group In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other... of a topological space.
1895	Henri Poincaré Henri Poincaré Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...	\| Simplicial homology Simplicial homology In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory.... .
1895	Henri Poincaré Henri Poincaré Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...	\| Fundamental work Analysis situs, the beginning of algebraic topology Algebraic topology Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology... .
c.1910	L. E. J. Brouwer	\| Brouwer develops intuitionism Intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism , is an approach to mathematics as the constructive mental activity of humans. That is, mathematics does not consist of analytic activities wherein deep properties of existence are revealed and applied... as a contribution to foundational debate in the period roughly 1910 to 1930 on mathematics, with intuitionistic logic Intuitionistic logic Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either... a by-product of an increasingly sterile discussion on formalism.
1923	Hermann Künneth	\| Künneth formula for homology of product of spaces.
1926	Heinrich Brandt Heinrich Brandt Heinrich Brandt was a German mathematician who was the first to develop the concept of groupoid....	\| defines the notion of groupoid Groupoid In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
1928	Arend Heyting Arend Heyting Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...	\| Brouwer's intuitionistic logic made into formal mathematics, as logic in which the Heyting algebra Heyting algebra In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b... replaces the Boolean algebra.
1929	Walther Mayer Walther Mayer Walter Mayer was an Austrian mathematician, born 1887 in Graz, Austria.Mayer, who was Jewish, studied at the Federal Institute of Technology in Zürich and the University of Paris before receiving his doctorate in 1912 from the University of Vienna...	\| Chain complex Chain complex In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or... es.
1930	Ernst Zermelo Ernst Zermelo Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated... –Abraham Fraenkel	\| Statement of the definitive ZF-axioms Zermelo–Fraenkel set theory In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes... of set theory, first stated in 1908 and improved upon since then.
c.1930	Emmy Noether Emmy Noether Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...	\| Module theory is developed by Noether and her students, and algebraic topology starts to be properly founded in abstract algebra Abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras... rather than by ad hoc arguments.
1932	Eduard Čech Eduard Cech Eduard Čech was a Czech mathematician born in Stračov, Bohemia . His research interests included projective differential geometry and topology. In 1921–1922 he collaborated with Guido Fubini in Turin...	\| Čech cohomology Cech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:... , homotopy groups of a topological space.
1933	Solomon Lefschetz Solomon Lefschetz Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...	\| Singular homology Singular homology In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n.... of topological spaces.
1934	Reinhold Baer Reinhold Baer Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....	\| Ext groups, Ext functor Ext functor In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :... (for abelian group Abelian group In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... s and with different notation).
1935	Witold Hurewicz Witold Hurewicz Witold Hurewicz was a Polish mathematician.- Early life and education :Witold Hurewicz was born to a Jewish family in Łódź, Russian Empire ....	\| Higher homotopy groups Homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space... of a topological space.
1936	Marshall Stone	\| Stone representation theorem for Boolean algebras initiates various Stone dualities Stone duality In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation... .
1937	Richard Brauer Richard Brauer Richard Dagobert Brauer was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory... –Cecil Nesbitt Cecil J. Nesbitt Cecil James Nesbitt, Ph.D., F.S.A., M.A.A.A. was a mathematician who was a Ph.D. student of Richard Brauer and wrote many influential papers in the early history of modular representation theory. He taught actuarial mathematics at the University of Michigan from 1938 to 1980. Nesbitt was born in...	\| Frobenius algebra Frobenius algebra In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in... s.
1938	Hassler Whitney Hassler Whitney Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...	\| "Modern" definition of cohomology Cohomology In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries... , summarizing the work since James Alexander James Waddell Alexander II James Waddell Alexander II was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others... and Andrey Kolmogorov Andrey Kolmogorov Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov... first defined cochains.
1940	Reinhold Baer Reinhold Baer Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....	\| Injective module Injective module In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers... s.
1940	Kurt Gödel Kurt Gödel Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the... –Paul Bernays Paul Bernays Paul Isaac Bernays was a Swiss mathematician, who made significant contributions to mathematical logic, axiomatic set theory, and the philosophy of mathematics. He was an assistant to, and close collaborator of, David Hilbert.-Biography:Bernays spent his childhood in Berlin. Bernays attended the...	\| Proper classes Class (set theory) In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context... in set theory.
1940	Heinz Hopf Heinz Hopf Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...	\| Hopf algebra Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally... s.
1941	Witold Hurewicz Witold Hurewicz Witold Hurewicz was a Polish mathematician.- Early life and education :Witold Hurewicz was born to a Jewish family in Łódź, Russian Empire ....	\| First fundamental theorem of homological algebra: Given a short exact sequence of spaces there exist a connecting homomorphism such that the long sequence of cohomology Cohomology In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries... groups of the spaces is exact.
1942	Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor... –Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| Universal coefficient theorem for Čech cohomology Cech cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:... ; later this became the general universal coefficient theorem Universal coefficient theorem In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A... . The notations Hom and Ext first appear in their paper.
1943	Norman Steenrod Norman Steenrod Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:...	\| Homology with local coefficients.
1943	Israel Gelfand Israel Gelfand Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand was a Soviet mathematician who made major contributions to many branches of mathematics, including group theory, representation theory and functional analysis... –Mark Naimark Mark Naimark Mark Aronovich Naimark was a Soviet mathematician.He was born in Odessa, Russian Empire into a Jewish family and died in Moscow, USSR...	\| Gelfand–Naimark theorem Gelfand–Naimark theorem In mathematics, the Gelfand–Naimark theorem states that an arbitrary C-algebra A is isometrically -isomorphic to a C-algebra of bounded operators on a Hilbert space... (sometimes called Gelfand isomorphism theorem): The category Haus of locally compact Hausdorff spaces with continuous proper maps as morphisms is equivalent to the category CAlg of commutative C-algebras with proper -homomorphisms as morphisms.
1944	Garrett Birkhoff Garrett Birkhoff Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father.... –Øystein Ore Øystein Ore Øystein Ore was a Norwegian mathematician.-Life:Ore was graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem...	\| Galois connection Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence... s generalizing the Galois correspondence: a pair of adjoint functors between two categories that arise from partially ordered sets (in modern formulation).
1944	Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...	\| "Modern" definition of singular homology Singular homology In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n.... and singular cohomology.
1945	Beno Eckmann	\| Defines the cohomology ring Cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is... building on Heinz Hopf Heinz Hopf Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age... 's work.

1945–1970

Year	Contributors	Event
1945	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:... –Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...	\| Start of category theory: axioms for categories Category (mathematics) In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose... , functors and natural transformations.
1945	Norman Steenrod Norman Steenrod Norman Earl Steenrod was a preeminent mathematician most widely known for his contributions to the field of algebraic topology.-Life:... –Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...	\| Eilenberg–Steenrod axioms for homology and cohomology.
1945	Jean Leray Jean Leray Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....	\| Starts sheaf theory: At this time a sheaf was a map assigned a module or a ring to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its p'th cohomology group.
1945	Jean Leray Jean Leray Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....	\| Defines Sheaf cohomology Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F... using his new concept of sheaf.
1946	Jean Leray Jean Leray Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....	\| Invents spectral sequences as a method for iteratively approximating cohomology groups by previous approximate cohomology groups. In the limiting case it gives the sought cohomology groups.
1948	Cartan seminar	\| Writes up sheaf theory for the first time.
1948	A. L. Blakers	\| Crossed complexes (called group systems by Blakers), after a suggestion of Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor... : A nonabelian generalizations of chain complex Chain complex In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or... es of abelian groups which are equivalent to strict ω-groupoids. They form a category Crs that has many satisfactory properties such as a monoidal structure Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... .
1949	John Henry Whitehead	\| Crossed module Crossed module In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H , and a homomorphism of groups... s.
1949	André Weil André Weil André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...	\| Formulates the Weil conjectures Weil conjectures In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields.... on remarkable relations between the cohomological structure of algebraic varieties over C and the diophantine structure of algebraic varieties over finite fields.
1950	Henri Cartan Henri Cartan Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...	\| In the book Sheaf theory from the Cartan seminar he defines: Sheaf space (étale space), support of sheaves axiomatically, sheaf cohomology Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F... with support in an axiomatic form and more.
1950	John Henry Whitehead	\| Outlines algebraic homotopy program for describing, understanding and calculating homotopy types of spaces and homotopy classes of mappings
1950	Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor... –Joe Zilber	\| Simplicial set Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... s as a purely algebraic model of well behaved topological spaces. A simplicial set can also be seen as a presheaf on the simplex category Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... . A category is a simplicial set such that the Segal maps are isomorphisms.
1951	Henri Cartan Henri Cartan Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...	\| Modern definition of sheaf theory in which a sheaf Sheaf (mathematics) In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of... is defined using open subsets instead of closed subsets of a topological space and all the open subsets are treated at once. A sheaf on a topological space X becomes a functor reminding of a function defined locally on X, and taking values in sets, abelian groups, commutative rings, modules or generally in any category C. In fact Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory... later made a dictionary between sheaves and functions. Another interpretation of sheaves is as continuously varying sets (a generalization of abstract sets). Its purpose is to provide a unified approach to connect local and global properties of topological spaces and to classify the obstructions for passing from local objects to global objects on a topological space by pasting together the local pieces. The C-valued sheaves on a topological space and their homomorphisms form a category.
1952	William Massey William S. Massey William Schumacher Massey is an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including Algebraic Topology .William Massey was...	\| Invents exact couples for calculating spectral sequences.
1953	Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...	\| Serre C-theory and Serre subcategories.
1955	Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...	\| Shows there is a 1-1 correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring (Serre–Swan theorem).
1955	Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...	\| Coherent sheaf cohomology in algebraic geometry.
1956	Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...	\| GAGA correspondence Algebraic geometry and analytic geometry In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several... .
1956	Henri Cartan Henri Cartan Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:... –Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...	\| Influential book: Homological Algebra, summarizing the state of the art in its topic at that time. The notation Tor_n and Extⁿ, as well as the concepts of projective module Projective module In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module... , projective and injective resolution of a module, derived functor Derived functor In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :... and hyperhomology Hyperhomology In homological algebra, the hyperhomology or hypercohomology of a complexof objects of an abelian category is an extension of the usual homology of an object to complexes.... appear in this book for the first time.
1956	Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...	\| Simplicial homotopy theory also called categorical homotopy theory: A homotopy theory completely internal to the category of simplicial sets.
1957	Charles Ehresmann Charles Ehresmann Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military... –Jean Bénabou	\| Pointless topology Pointless topology In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann... building on Marshall Stone's work.
1957	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Abelian categories in homological algebra that combine exactness and linearity.
1957	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Influential Tohoku paper rewrites homological algebra Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and... ; proving Grothendieck duality (Serre duality for possibly singular algebraic varieties). He also showed that the conceptual basis for homological algebra over a ring also holds for linear objects varying as sheaves over a space.
1957	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Grothendieck relative point of view Grothendieck's relative point of view Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object... , S-schemes.
1957	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Grothendieck–Hirzebruch–Riemann–Roch theorem Grothendieck–Hirzebruch–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem... for smooth schemes; the proof introduces K-theory K-theory In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It... .
1957	Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...	\| Kan complexes: Simplicial set Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... s (in which every horn has a filler) that are geometric models of simplicial ∞-groupoids. Kan complexes are also the fibrant (and cofibrant) objects of model categories Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... of simplicial sets for which the fibrations are Kan fibration Kan fibration In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category for simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category... s.
1958	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Starts new foundation of algebraic geometry Algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex... by generalizing varieties and other spaces in algebraic geometry to schemes Scheme (mathematics) In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern... which have the structure of a category with open subsets as objects and restrictions as morphisms. Schemes forma a category that is a Grothendieck topos, and to a scheme and even a stack one may associate a Zariski topos, an étale topos, a fppf topos, a fpqc topos, a Nisnevich topos, a flat topos, ... depending on the topology imposed on the scheme. The whole of algebraic geometry was categorized with time.
1958	Roger Godement Roger Godement Roger Godement is a French mathematician, known for his work in functional analysis, and also his expository books.He started as a student at the École normale supérieure in 1940, where he became a student of Henri Cartan...	\| Monads Monad (category theory) In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations... in category theory (then called standard constructions and triples). Monads generalize classical notions from universal algebra Universal algebra Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures.... and can in this sense be thought of as an algebraic theory Theory (mathematical logic) In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom... over a category: the theory of the category of T-algebras. An algebra for a monad subsumes and generalizes the notion of a model for an algebraic theory.
1958	Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...	\| Adjoint functors Adjoint functors In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency... .
1958	Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...	\| Limits Limit (category theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits.... in category theory.
1958	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Fibred categories Fibred category Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined... .
1959	Bernard Dwork Bernard Dwork Bernard Morris Dwork was an American mathematician, known for his application of p-adic analysis to local zeta functions, and in particular for the first general results on the Weil conjectures. Together with Kenkichi Iwasawa he received the Cole Prize in 1962.Dwork received his Ph.D. at Columbia...	\| Proves the rationality part of the Weil conjectures Weil conjectures In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields.... (the first conjecture).
1959	Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...	\| Algebraic K-theory Algebraic K-theory In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n.... launched by explicit analogy of ring theory Ring theory In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers... with geometric cases.
1960	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Fiber functors
1960	Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an...	\| Kan extension Kan extension Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M... s
1960	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Formal algebraic geometry and formal scheme Formal scheme In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme... s
1960	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Representable functor Representable functor In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a... s
1960	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Categorizes Galois theory (Grothendieck galois theory Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry... )
1960	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Descent theory Descent (category theory) In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated... : An idea extending the notion of gluing in topology to schemes Scheme (mathematics) In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern... to get around the brute equivalence relations. It also generalizes localization Localization of a topological space In mathematics, well behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in .... in topology
1961	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Local cohomology Local cohomology In mathematics, local cohomology is a chapter of homological algebra and sheaf theory introduced into algebraic geometry by Alexander Grothendieck. He developed it in seminars in 1961 at Harvard University, and 1961-2 at IHES. It was later written up as SGA2... . Introduced at a seminar in 1961 but the notes are published in 1967
1961	Jim Stasheff	\| Associahedra Associahedron In mathematics, an associahedron or Stasheff polytope Kn is a convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a word of n letters and the edges correspond to single application of the associativity rule.Initially Jim Stasheff... later used in the definition of weak n-categories Weak n-category In category theory, weak n-categories are a generalization of the notion of n-category where composition is not strictly associative but only associative up to coherent equivalence. There is currently much work to determine what the coherence laws should be for those. Weak n-categories have become...
1961	Richard Swan Richard Swan Richard Gordon Swan is an American mathematician who is best known for Swan's theorem. His work has mainly been in the area of algebraic K-theory.-External links:**...	\| Shows there is a 1-1 correspondence between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring C(X) of continuous functions on X (Serre–Swan theorem)
1963	Frank Adams–Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| PROP categories and PACT categories for higher homotopies. PROPs are categories for describing families of operations with any number of inputs and outputs. Operads are special PROPs with operations with only one output
1963	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Étale topology Étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic... , a special Grothendieck topology on schemes
1963	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Étale cohomology Étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
1963	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Grothendieck toposes, which are categories which are like universes (generalized spaces) of sets in which one can do mathematics
1963	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Algebraic theories Algebraic theory In mathematical logic, an algebraic theory is one that uses axioms stated entirely in terms of equations between terms with free variables. Inequalities and quantifiers are specifically disallowed. Sentential logic is the subset of first-order logic involving only algebraic sentences.Saying that... and algebraic categories
1963	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Founds Categorical logic Categorical logic Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor... , discover internal logics of categories and recognizes its importance and introduces Lawvere theories. Essentially categorical logic is a lift of different logics to being internal logics of categories. Each kind of category with extra structure corresponds to a system of logic with its own inference rules. A Lawvere theory is an algebraic theory Theory (mathematical logic) In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom... as a category with finite products and possessing a "generic algebra" (a generic group). The structures described by a Lawvere theory are models of the Lawvere theory
1963	Jean-Louis Verdier Jean-Louis Verdier Jean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...	\| Triangulated categories and triangulated functors. Derived categories Derived category In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C... and derived functor Derived functor In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :... s are special cases of these
1963	Jim Stasheff	\| A_∞-algebras: dg-algebra analogs of topological monoids associative up to homotopy appearing in topology (i.e. H-space H-space In mathematics, an H-space is a topological space X together with a continuous map μ : X × X → X with an identity element e so that μ = μ = x for all x in X... s)
1963	Jean Giraud Jean Giraud (mathematician) Jean Giraud was a French mathematician, a student of Alexander Grothendieck and the author of the book "Cohomologie non abélienne" ....	\| Giraud characterization theorem characterizing Grothendieck toposes as categories of sheaves over a small site
1963	Charles Ehresmann Charles Ehresmann Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...	\| Internal category theory: Internalization of categories in a category V with pullbacks is replacing the category Set (same for classes instead of sets) by V in the definition of a category. Internalization is a way to rise the categorical dimension
1963	Charles Ehresmann Charles Ehresmann Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...	\| Multiple categories and multiple functors
1963	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| Monoidal categories also called tensor categories: Strict 2-categories with one object made by a relabelling trick to categories with a tensor product Tensor product In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general... of objects that is secretly the composition of morphisms in the 2-category. There are several object in a monoidal category since the relabelling trick makes 2-morphisms of the 2-category to morphisms, morphisms of the 2-category to objects and forgets about the single object. In general a higher relabelling trick works for n-categories N-category In mathematics, n-categories are a high-order generalization of the notion of category. The category of n-categories n-Cat is defined by induction on n by:... with one object to make general monoidal categories. The most common examples include: ribbon categories, braided tensor categories, spherical categories, compact closed categories Compact closed category In category theory, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space... , symmetric tensor categories, modular categories, autonomous categories Autonomous category In mathematics, an autonomous category is a monoidal category where dual objects exist.-Definition:A left autonomous category is a monoidal category where every object has a left dual. An autonomous category is a monoidal category where every object has both a left and a right dual... , categories with duality
1963	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| Mac Lane coherence theorem Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... for determining commutativity of diagrams in monoidal categories
1964	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| ETCS Elementary Theory of the Category of Sets: An axiomatization of the category of sets Category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B... which is also the constant case of an elementary topos
1964	Barry Mitchell–Peter Freyd	\| Mitchell–Freyd embedding theorem Mitchell's embedding theorem Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem, is a result stating that every abelian category admits a full and exact embedding into the category of R-modules... : Every small abelian category Abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative... admits an exact and full embedding into the category of (left) modules Mod_R over some ring R
1964	Rudolf Haag Rudolf Haag Rudolf Haag is a German physicist. He is best known for his contributions to the algebraic formulation of axiomatic quantum field theory, namely the Haag-Kastler axioms... –Daniel Kastler Daniel Kastler Daniel Kastler is a French theoretical physicist, working at University of Aix-Marseille on non-commutative geometry.He is best known for his 1964 article with Rudolf Haag on algebraic quantum field theory, which is one of the ....	\| Algebraic quantum field theory after ideas of Irving Segal Irving Segal Irving Ezra Segal was a mathematician known for work on theoretical quantum mechanics.He was at the Massachusetts Institute of Technology...
1964	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Topologizes categories axiomatically by imposing a Grothendieck topology Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion... on categories which are then called sites Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion... . The purpose of sites is to define coverings on them so sheaves over sites can be defined. The other "spaces" one can define sheaves for except topological spaces are locales
1964	Michael Artin Michael Artin Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,... –Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| ℓ-adic cohomology, technical development in SGA4 of the long-anticipated Weil cohomology.
1964	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Proves the Weil conjectures Weil conjectures In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields.... except the analogue of the Riemann hypothesis
1964	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Six operations formalism in homological algebra Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and... ; Rf_*, f⁻¹, Rf_!, f^!, ⊗^L, RHom, and proof of its closedness
1964	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Introduced in a letter to Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:... conjectural motives (algebraic geometry) Motive (algebraic geometry) In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer... to express the idea that there is a single universal cohomology theory underlying the various cohomology theories for algebraic varieties. According to Grothendiecks philosophy there should be a universal cohomology functor attaching a pure motive h(X) to each smooth projective variety X. When X is not smooth or projective h(X) must be replaced by a more general mixed motive which has a weight filtration whose quotients are pure motivess. The category of motives Motive (algebraic geometry) In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer... (the categorical framework for the universal cohomology theory) may be used as an abstract substitute for singular cohomology (and rational cohomology) to compare, relate and unite "motivated" properties and parallel phenomena of the various cohomology theories and to detect topological structure of algebraic varieties. The categories of pure motives and of mixed motives are abelian tensor categories and the category of pure motives is also a Tannakian category Tannakian category In mathematics, a tannakian category is a particular kind of monoidal category C, equipped with some extra structure relative to a given field K. The role of such categories C is to approximate, in some sense, the category of linear representations of an algebraic group G defined over K... . Categories of motives are made by replacing the category of varieties by a category with the same objects but whose morphisms are correspondences, modulo a suitable equivalence relation. Different equivalences Equivalence relation In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell... give different theories. Rational equivalence gives the category of Chow motives with Chow groups as morphisms which are in some sense universal. Every geometric cohomology theory is a functor on the category of motives. Each induced functor ρ:motives modulo numerical equivalence→graded Q-vector spaces is called a realization Realization (probability) In probability and statistics, a realization, or observed value, of a random variable is the value that is actually observed . The random variable itself should be thought of as the process how the observation comes about... of the category of motives, the inverse functors are called improvements. Mixed motives explain phenomena in as diverse areas as: Hodge theory, algebraic K-theory, polylogarithms, regulator maps, automorphic forms, L-functions, ℓ-adic representations, trigonometric sums, homotopy of algebraic varieties, algebraic cycles, moduli spaces and thus has the potential of enriching each area and of unifying them all.
1965	Edgar Brown	Abstract homotopy categories: A proper framework for the study of homotopy theory of CW-complexes
1965	Max Kelly Max Kelly Gregory Maxwell Kelly, 1930-2007, mathematician, founded the thriving Australian school of category theory.A native of Australia, Kelly obtained his Ph.D. at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory...	\| dg-categories Differential graded category In mathematics, especially homological algebra, a differential graded category or DG category for short, is a category whose morphism sets are endowed with the additional structure of a differential graded Z-module....
1965	Max Kelly Max Kelly Gregory Maxwell Kelly, 1930-2007, mathematician, founded the thriving Australian school of category theory.A native of Australia, Kelly obtained his Ph.D. at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory... –Samuel Eilenberg Samuel Eilenberg Samuel Eilenberg was a Polish and American mathematician of Jewish descent. He was born in Warsaw, Russian Empire and died in New York City, USA, where he had spent much of his career as a professor at Columbia University.He earned his Ph.D. from University of Warsaw in 1936. His thesis advisor...	\| Enriched category theory: Categories C enriched over a category V are categories with Hom-sets Hom_C not just a set or class but with the structure of objects in the category V. Enrichment over V is a way to rise the categorical dimension
1965	Charles Ehresmann Charles Ehresmann Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...	\| Defines both strict 2-categories 2-category In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category... and strict n-categories N-category In mathematics, n-categories are a high-order generalization of the notion of category. The category of n-categories n-Cat is defined by induction on n by:...
1966	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Crystals (a kind of sheaf used in crystalline cohomology Crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field.... )
1966	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| ETAC Elementary theory of abstract categories, first proposed axioms for Cat or category theory using first order logic
1967	Jean Bénabou	\| Bicategories Bicategory In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B... (weak 2-categories) and weak 2-functors
1967	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Founds synthetic differential geometry Synthetic differential geometry In mathematics, synthetic differential geometry is a reformulation of differential geometry in the language of topos theory, in the context of an intuitionistic logic characterized by a rejection of the law of excluded middle. There are several insights that allow for such a reformulation...
1967	Simon Kochen–Ernst Specker	\| Kochen–Specker theorem in quantum mechanics
1967	Jean-Louis Verdier Jean-Louis Verdier Jean-Louis Verdier was a French mathematician who worked, under the guidance of Alexander Grothendieck, on derived categories and Verdier duality...	\| Defines derived categories and redefines derived functors in terms of derived categories
1967	Peter Gabriel–Michel Zisman	\| Axiomatizes simplicial homotopy theory
1967	Daniel Quillen	\| Quillen Model categories Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... and Quillen model functors Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... : A framework for doing homotopy theory in an axiomatic way in categories and an abstraction of homotopy categories in such a way that hC = C[W⁻¹] where W⁻¹ are the inverted weak equivalence Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... s of the Quillen model category C. Quillen model categories are homotopically complete and cocomplete, and come with a built-in Eckmann–Hilton duality Eckmann–Hilton duality In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category.It...
1967	Daniel Quillen	\| Homotopical algebra Homotopical algebra In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases... (published as a book and also sometimes called noncommutative homological algebra): The study of various model categories Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... and the interplay between fibrations, cofibrations and weak equivalences in arbitrary closed model categories
1967	Daniel Quillen	\| Quillen axioms for homotopy theory in model categories Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
1967	Daniel Quillen	\| First fundamental theorem of simplicial homotopy theory: The category of simplicial sets is a (proper) closed (simplicial) model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
1967	Daniel Quillen	\| Second fundamental theorem of simplicial homotopy theory: The realization functor and the singular functor is an equivalence of categories hΔ and hTop (Δ the category of simplicial sets)
1967	Jean Bénabou	\| V-actegories: A category C with an action ⊗ :V × C → C which is associative and unital up to coherent isomorphism, for V a symmetric monoidal category. V-actegories can be seen as the categorification of R-modules over a commutative ring R
1968	Chen Yang-Rodney Baxter	\| Yang-Baxter equation Yang-Baxter equation The Yang–Baxter equation is an equation which was first introduced in the field of statistical mechanics. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1971... , later used as a relation in braided monoidal categories for crossings of braids
1968	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Crystalline cohomology Crystalline cohomology In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by and developed by . Its values are modules over rings of Witt vectors over the base field.... : A p-adic cohomology theory in characteristic p invented to fill the gap left by étale cohomology Étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures... which is deficient in using mod p coefficients for this case. It is sometimes referred to by Grothendieck as the yoga of de Rham coefficients and Hodge coefficients since crystalline cohomology of a variety X in characteristic p is like de Rham cohomology De Rham cohomology In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes... mod p of X and there is an isomorphism between de Rham cohomology groups and Hodge cohomology groups of harmonic forms
1968	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Grothendieck connection Grothendieck connection In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.-Introduction and motivation:...
1968	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Formulates the standard conjectures on algebraic cycles Standard conjectures on algebraic cycles In mathematics, the standard conjectures about algebraic cycles is a package of several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Grothendieck, was to prove that his...
1968	Michael Artin Michael Artin Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...	\| Algebraic space Algebraic space In mathematics, an algebraic space is a generalization of the schemes of algebraic geometry introduced by Michael Artin for use in deformation theory... s in algebraic geometry as a generalization of schemes
1968	Charles Ehresmann Charles Ehresmann Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...	\| Sketches (category theory): An alternative way of presenting a theory (which is categorical in character as opposed to linguistic) whose models are to study in appropriate categories. A sketch is a small category with a set of distinguished cones and a set of distinguished cocones satisfying some axioms. A model of a sketch is a set-valued functor transforming the distinguished cones into limit cones and the distinguished cocones into colimit cones. The categories of models of sketches are exactly the accessible categories Accessible category The theory of accessible categories was introduced in 1989 by mathematicians Michael Makkai and Robert Paré in the setting of category theory. While the original motivation came from model theory, a branch of mathematical logic, it turned out that accessible categories have applications in homotopy...
1968	Joachim Lambek Joachim Lambek Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :...	\| Multicategories Multicategory In mathematics , a multicategory is a generalization of the concept of category that allows morphisms of multiple arity...
1969	Max Kelly Max Kelly Gregory Maxwell Kelly, 1930-2007, mathematician, founded the thriving Australian school of category theory.A native of Australia, Kelly obtained his Ph.D. at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory... -Nobuo Yoneda Nobuo Yoneda was a Japanese mathematician and computer scientist. The Yoneda lemma in category theory is named after him. In computer science, he is known for his work on ALGOL dialects.-References:...	\| Ends and coends
1969	Pierre Deligne Pierre Deligne - See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :... -David Mumford David Mumford David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...	\| Deligne-Mumford stacks as a generalization of schemes Scheme (mathematics) In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
1969	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Doctrines (category theory), a doctrine is a monad on a 2-category
1970	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:... -Myles Tierney	\| Elementary toposes: Categories modeled after the category of sets Category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B... which are like universe Universe (mathematics) In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation... s (generalized spaces) of sets in which one can do mathematics. One of many ways to define a topos is: a properly cartesian closed category Cartesian closed category In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in... with a subobject classifier Subobject classifier In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements... . Every Grothendieck topos is an elementary topos
1970	John Conway John Horton Conway John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...	\| Skein theory of knots: The computation of knot invariants by skein modules. Skein modules can be based on quantum invariant Quantum invariant In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.-List of invariants:Finite type invariantKontsevich invariant*Kashaev's invariant... s

1971–1980

Year	Contributors	Event
1971	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| Influential book: Categories for the working mathematician, which became the standard reference in category theory
1971	Horst Herrlich-Oswald Wyler	\| Categorical topology: The study of topological categories Topological category In mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory .... of structured sets (generalizations of topological spaces, uniform spaces and the various other spaces in topology) and relations between them, culminating in universal topology. General categorical topology study and uses structured sets in a topological category as general topology study and uses topological spaces. Algebraic categorical topology tries to apply the machinery of algebraic topology for topological spaces to structured sets in a topological category.
1971	Harold Temperley Harold Neville Vazeille Temperley Harold Neville Vazeille Temperley is a mathematical physicist working on "lattice gases". His father, Harold William Vazeille Temperley, was a distinguished British historian.... -Elliott Lieb	\| Temperley–Lieb algebras: Algebras of tangles defined by generators of tangles and relations among them
1971	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:... –Myles Tierney	\| Lawvere–Tierney topology on a topos
1971	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:... –Myles Tierney	\| Topos theoretic forcing (forcing in toposes): Categorization of the set theoretic forcing Forcing (mathematics) In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory... method to toposes for attempts to prove or disprove the continuum hypothesis Continuum hypothesis In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900... , independence of the axiom of choice, etc. in toposes
1971	Bob Walters-Ross Street	\| Yoneda structures on 2-categories
1971	Roger Penrose Roger Penrose Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...	\| String diagram String diagram In category theory, string diagrams are a way of representing 2-cells in 2-categories. The idea is to represent structures of dimension d by structures of dimension 2-d, using the Poincaré duality... s to manipulate morphisms in a monoidal category
1971	Jean Giraud Jean Giraud (mathematician) Jean Giraud was a French mathematician, a student of Alexander Grothendieck and the author of the book "Cohomologie non abélienne" ....	\| Gerbe Gerbe In mathematics, a gerbe is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as a generalization of principal bundles to the setting of 2-categories... s: Categorified principal bundles that are also special cases of stacks
1971	Joachim Lambek Joachim Lambek Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :...	\| Generalizes the Haskell-Curry-William-Howard correspondence to a three way isomorphism between types, propositions and objects of a cartesian closed category
1972	Max Kelly Max Kelly Gregory Maxwell Kelly, 1930-2007, mathematician, founded the thriving Australian school of category theory.A native of Australia, Kelly obtained his Ph.D. at Cambridge University in homological algebra in 1957, publishing his first paper in that area in 1959, Single-space axioms for homology theory...	\| Clubs (category theory) and coherence (category theory). A club is a special kind of 2-dimensional theory or a monoid in Cat/(category of finite sets and permutations P), each club giving a 2-monad on Cat
1972	John Isbell	\| Locales: A "generalized topological space" or "pointless spaces" defined by a lattice (a complete Heyting algebra Heyting algebra In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b... also called a Brouwer lattice) just as for a topological space the open subsets form a lattice. If the lattice possess enough points it is a topological space. Locales are the main objects of pointless topology Pointless topology In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann... , the dual objects being frames. Both locales and frames form categories that are each others opposite. Sheaves can be defined over locales. The other "spaces" one can define sheaves over are sites. Although locales were known earlier John Isbell first named them
1972	Ross Street	\| Formal theory of monads: The theory of monads Monad (category theory) In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations... in 2-categories
1972	Peter Freyd	\| Fundamental theorem of topos theory: Every slice category (E,Y) of a topos E is a topos and the functor f*:(E,X)→(E,Y) preserves exponentials and the subobject classifier object Ω and has a right and left adjoint functor
1972	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Universes (mathematics) Universe (mathematics) In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation... for sets
1972	Jean Bénabou–Ross Street	\| Cosmoses (category theory) which categorize universe Universe (mathematics) In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation... s: A cosmos is a generalized universe of 1-categories in which you can do category theory. When set theory is generalized to the study of a Grothendieck topos, the analogous generalization of category theory is the study of a cosmos. Ross Street definition: A bicategory Bicategory In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou.Formally, a bicategory B... such that 1) small bicoproducts exist 2) each monad Monad (category theory) In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations... admits a Kleisli construction (analogous to the quotient of an equivalence relation in a topos) 3) it is locally small-cocomplete 4) there exists a small Cauchy generator Generator (category theory) In category theory in mathematics a generator of a category \mathcal C is an object G of the category, such that for any two different morphisms f, g: X \rightarrow Y in \mathcal C, there is a morphism h : G \rightarrow X, such that the compositions f \circ h \neq g \circ h.Generators are central... . Cosmoses are closed under dualization, parametrization and localization. Ross Street also introduces elementary cosmoses. Jean Bénabou definition: A bicomplete symmetric monoidal closed category
1972	Peter May J. Peter May Jon Peter May is an American mathematician, working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra. He is known, in particular, for inventing the term operads and the May spectral sequence.He received a B.A. from Swarthmore College in...	\| Operads: An abstraction of the family of composable functions of several variables together with an action of permutation of variables. Operads can be seen as algebraic theories and algebras over operads are then models of the theories. Each operad gives a monad Monad (category theory) In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations... on Top. Multicategories with one object are operads. PROPs generalize operads to admit operations with several inputs and several outputs. Operads are used in defining opetopes, higher category theory, homotopy theory, homological algebra, algebraic geometry, string theory and many other areas.
1972	William Mitchell-Jean Bénabou	\| Mitchell-Bénabou internal language of a topos Topos In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space... es: For a topos E with subobject classifier Subobject classifier In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. As the name suggests, what a subobject classifier does is to identify/classify subobjects of a given object according to which elements... object Ω a language (or type theory Type theory In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general... ) L(E) where: 1) the types are the objects of E 2) terms of type X in the variables x_i of type X_i are polynomial expressions φ(x₁,...,x_m):1→X in the arrows x_i:1→X_i in E 3) formulas are terms of type Ω (arrows from types to Ω) 4) connectives are induced from the internal Heyting algebra Heyting algebra In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b... structure of Ω 5) quantifiers bounded by types and applied to formulas are also treated 6) for each type X there are also two binary relations =_X (defined applying the diagonal map to the product term of the arguments) and ∈_X (defined applying the evaluation map to the product of the term and the power term of the arguments). A formula is true if the arrow which interprets it factor through the arrow true:1→Ω. The Mitchell-Bénabou internal language is a powerful way to describe various objects in a topos as if they were sets and hence is a way of making the topos into a generalized set theory, to write and prove statements in a topos using first order intuitionistic predicate logic, to consider toposes as type theories and to express properties of a topos. Any language L also generates a linguistic topos E(L)
1973	Chris Reedy	\| Reedy categories: Categories of "shapes" that can be used to do homotopy theory. A Reedy category is a category R equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape R. The most important consequence of a Reedy structure on R is the existence of a model structure on the functor category Functor category In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors... M^R whenever M is a model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... . Another advantage of the Reedy structure is that its cofibrations, fibrations and factorizations are explicit. In a Reedy category there is a notion of an injective and a surjective morphism such that any morphism can be factored uniquely as a surjection followed by an injection. Examples are the ordinal α considered as a poset and hence a category. The opposite R° of a Reedy category R is a Reedy category. The simplex category Δ and more generally for any simplicial set Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... X its category of simplices Δ/X is a Reedy category. The model structure on M^Δ for a model category M is described in an unpublished manuscript by Chris Reedy
1973	Kenneth Brown–Stephen Gersten	\| Shows the existence of a global closed model structure Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... on the categegory of simplicial sheaves on a topological space, with weak assumptions on the topological space
1973	Kenneth Brown	\| Generalized sheaf cohomology of a topological space X with coefficients a sheaf on X with values in Kans category of spectra Spectrum (homotopy theory) In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory.... with some finiteness conditions. It generalizes generalized cohomology theory and sheaf cohomology Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F... with coefficients in a complex of abelian sheaves
1973	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Finds that Cauchy completeness can be expressed for general enriched categories Enriched category In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely... with the category of generalized metric spaces as a special case. Cauchy sequences become left adjoint modules and convergence become representability
1973	Jean Bénabou	\| Distributors Profunctor In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. They are related to the notion of correspondences.- Definition :... (also called modules, profunctors, directed bridges)
1973	Pierre Deligne Pierre Deligne - See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...	\| Proves the last of the Weil conjectures Weil conjectures In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields.... , the analogue of the Riemann hypothesis
1973	John Boardman-Rainer Vogt	\| Segal categories: Simplicial analogues of A_∞-categories. They naturally generalize simplicial categories, in that they can be regarded as simplicial categories with composition only given up to homotopy. Def: A simplicial space X such that X₀ (the set of points) is a discrete simplicial set Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... and the Segal map φ_k : X_k → X₁ × _{X ₀} ... × _{X ₀}X₁ (induced by X(α_i):X_k → X₁) assigned to X is a weak equivalence of simplicial sets for k≥2. Segal categories are a weak form of S-categories, in which composition is only defined up to a coherent system of equivalences. Segal categories were defined one year later implicitly by Graeme Segal Graeme Segal Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil... . They were named Segal categories first by William Dwyer–Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an... –Jeffrey Smith 1989. In their famous book Homotopy invariant algebraic structures on topological spaces John Boardman and Rainer Vogt called them quasi-categories Quasi-category In mathematics, a quasi-category is a higher categorical generalization of a notion of a category introduced by .André Joyal has much advanced the study of quasi-categories showing that most of the usual basic category... . A quasi-category is a simplicial set satisfying the weak Kan condition, so quasi-categories are also called weak Kan complexes
1973	Daniel Quillen	\| Frobenius categories: An exact category Exact category In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition... in which the classes of injective and projective objects coincide and for all objects x in the category there is a deflation P(x)→x (the projective cover of x) and an inflation x→I(x) (the injective hull of x) such that both P(x) and I(x) are in the category of pro/injective objects. A Frobenius category E is an example of a model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... and the quotient E/P (P is the class of projective/injective objects) is its homotopy category hE
1974	Michael Artin Michael Artin Michael Artin is an American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry. and also generally recognized as one of the outstanding professors in his field.Artin was born in Hamburg,...	\| Generalizes Deligne–Mumford stacks to Artin stacks
1974	Robert Paré	\| Paré monadicity theorem: E is a topos→E° is monadic over E
1974	Andy Magid	\| Generalizes Grothendiecks Galois theory Grothendieck's Galois theory In mathematics, Grothendieck's Galois theory is a highly abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry... from groups to the case of rings using Galois groupoids
1974	Jean Bénabou	\| Logic of fibred categories Fibred category Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined...
1974	John Gray	\| Gray categories with Gray tensor product
1974	Kenneth Brown	\| Writes a very influential paper that defines Browns categories of fibrant objects and dually Brown categories of cofibrant objects
1974	Shiing-Shen Chern Shiing-Shen Chern Shiing-Shen Chern was a Chinese American mathematician, one of the leaders in differential geometry of the twentieth century.-Early years in China:... –James Simons	\| Chern–Simons theory: A particular TQFT which describe knot and manifold invariants, at that time only in 3D
1975	Saul Kripke Saul Kripke Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center... –André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :...	\| Kripke–Joyal semantics of the Mitchell–Bénabou internal language for toposes: The logic in categories of sheaves is first order intuitionistic predicate logic
1975	Radu Diaconescu	\| Diaconescu theorem: The internal axiom of choice holds in a topos Topos In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space... → the topos is a boolean topos. So in IZF the axiom of choice implies the law of excluded middle
1975	Manfred Szabo	\| Polycategories
1975	William Lawvere William Lawvere Francis William Lawvere is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.-Biography:...	\| Observes that Delignes theorem about enough points in a coherent topos implies the Gödel completeness theorem Gödel's completeness theorem Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929.... for first order logic in that topos
1976	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Schematic homotopy types
1976	Marcel Crabbe	\| Heyting categories also called logoses: Regular categories Regular category In category theory, a regular category is a category with finite limits and coequalizers of kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity... in which the subobjects of an object form a lattice, and in which each inverse image map has a right adjoint. More precisely a coherent category C such that for all morphisms f:A→B in C the functor f*:Sub_C(B)→Sub_C(A) has a left adjoint and a right adjoint. Sub_C(A) is the preorder Preorder In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders... of subobjects of A (the full subcategory of C/A whose objects are subobjects of A) in C. Every topos Topos In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space... is a logos. Heyting categories generalize Heyting algebra Heyting algebra In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b... s.
1976	Ross Street	\| Computads
1977	Michael Makkai Michael Makkai Michael Makkai is a Canadian mathematician, specializing in mathematical logic. He works in model theory, category theory, algebraic logic, and in the theory of topoi. He graduated from the Eötvös Loránd University, Budapest, then worked at the Mathematical Institute of the Hungarian Academy of... –Gonzalo Reyes	\| Develops the Mitchell–Bénabou internal language of a topos thoroughly in a more general setting
1977	Andre Boileau–André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... –Jon Zangwill	\| LST Local set theory: Local set theory is a typed set theory whose underlying logic is higher order intuitionistic logic Intuitionistic logic Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either... . It is a generalization of classical set theory, in which sets are replaced by terms of certain types. The category C(S) built out of a local theory S whose objects are the local sets (or S-sets) and whose arrows are the local maps (or S-maps) is a linguistic topos. Every topos E is equivalent to a linguistic topos C(S(E))
1977	John Roberts	\| Introduces most general nonabelian cohomology of ω-categories with ω-categories as coefficients when he realized that general cohomology is about coloring simplices in ω-categories. There are two methods of constructing general nonabelian cohomology, as nonabelian sheaf cohomology in terms of descent Descent (category theory) In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated... for ω-category valued sheaves, and in terms of homotopical cohomology theory which realizes the cocycles. The two approaches are related by codescent Descent (category theory) In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
1978	John Roberts	\| Complicial sets (simplicial sets with structure or enchantment)
1978	Francois Bayen–Moshe Flato–Chris Fronsdal–Andre Lichnerowicz André Lichnerowicz André Lichnerowicz was a noted French differential geometer and mathematical physicist of Polish descent.-Biography:... –Daniel Sternheimer	\| Deformation quantization, later to be a part of categorical quantization
1978	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :...	\| Combinatorial species Combinatorial species In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures in terms of generating functions. Examples of discrete structures are graphs, permutations, trees, and so on; each of these has an associated generating function... in enumerative combinatorics Enumerative combinatorics Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations...
1978	Don Anderson	\| Building on work of Kenneth Brown defines ABC (co)fibration categories for doing homotopy theory and more general ABC model categories, but the theory lies dormant until 2003. Every Quillen model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... is an ABC model category. A difference to Quillen model categories is that in ABC model categories fibrations and cofibrations are independent and that for an ABC model category M^D is an ABC model category. To a ABC (co)fibration category is canonically associated a (left) right Heller derivator Derivator In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories and provide at the same time a language for homotopical algebra.... . Topological spaces with homotopy equivalences as weak equivalences, Hurewicz cofibrations as cofibrations and Hurewicz fibrations as fibrations form an ABC model category, the Hurewicz model structure on Top. Complexes of objects in an abelian category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations form an ABC precofibration category
1979	Don Anderson	\| Anderson axioms for homotopy theory in categories with a fraction functor
1980	Alexander Zamolodchikov Alexander Zamolodchikov Alexander Borissowitsch Zamolodchikov is a Russian physicist, known for his contributions to condensed matter physics and string theory.Born near Dubna,...	\| Zamolodchikov equation also called tetrahedron equation
1980	Ross Street	\| Bicategorical Yoneda lemma Yoneda lemma In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory...
1980	Masaki Kashiwara Masaki Kashiwara is a Japanese mathematician. He was a student of Mikio Sato at the University of Tokyo. Sato and Kashiwara have collaborated on algebraic analysis and D-module theory.He is a member of the French Academy of Sciences.- Concepts named after Kashiwara :... –Zoghman Mebkhout	\| Proves the Riemann–Hilbert correspondence Riemann–Hilbert correspondence In mathematics, the Riemann–Hilbert correspondence is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups... for complex manifolds
1980	Peter Freyd	\| Numerals in a topos

1981–1990

Year	Contributors	Event
1981	Shigeru Mukai	\| Mukai–Fourier transform
1982	Bob Walters	\| Enriched categories Enriched category In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely... with bicategories as a base
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Pursuing stacks: Manuscript circulated from Bangor, written in English in response to a correspondence in English with Ronald Brown Ronald Brown (mathematician) Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:... and Tim Porter, starting with a letter addressed to Daniel Quillen, developing mathematical visions in a 629 pages manuscript, a kind of diary, and to be published by the Société Mathématique de France, edited by G. Maltsiniotis.
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| First appearance of strict ∞-categories in pursuing stacks, following a 1981 published definition by Ronald Brown Ronald Brown (mathematician) Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:... and Philip J. Higgins.
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Fundamental infinity groupoid: A complete homotopy invariant Π_∞(X) for CW-complexes X. The inverse functor is the geometric realization functor \|.\| and together they form an "equivalence" between the category of CW-complexes and the category of ω-groupoids
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Homotopy hypothesis: The homotopy category of CW-complexes is Quillen equivalent Quillen adjunction In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho and Ho via the total derived functor construction... to a homotopy category of reasonable weak ∞-groupoids
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Grothendieck derivators Derivator In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories and provide at the same time a language for homotopical algebra.... : A model for homotopy theory similar to Quilen model categories Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... but more satisfactory. Grothendieck derivators are dual to Heller derivators Derivator In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories and provide at the same time a language for homotopical algebra....
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Elementary modelizers: Categories of presheaves that modelize homotopy types (thus generalizing the theory of simplicial set Simplicial set In mathematics, a simplicial set is a construction in categorical homotopy theory which is a purely algebraic model of the notion of a "well-behaved" topological space... s). Canonical modelizers are also used in pursuing stacks
1983	Alexander Grothendieck Alexander Grothendieck Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...	\| Smooth functor Smooth functor In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense that it sends smoothly parameterized families of vector spaces to smoothly parameterized families of... s and proper functors
1984	Vladimir Bazhanov–Razumov Stroganov	\| Bazhanov–Stroganov d-simplex equation generalizing the Yang–Baxter equation and the Zamolodchikov equation
1984	Horst Herrlich	\| Universal topology in categorical topology: A unifying categorical approach to the different structured sets (topological structures such as topological spaces and uniform spaces) whose class form a topological category similar as universal algebra is for algebraic structures
1984	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :...	\| Simplicial sheaves (sheaves with values in simplicial sets). Simplicial sheaves on a topological space X is a model for the hypercomplete ∞-topos Sh(X)^{^}
1984	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :...	\| Shows that the category of simplicial objects in a Grothendieck topos has a closed model structure Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...
1984	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... –Myles Tierney	\| Main Galois theorem for toposes: Every topos is equivalent to a category of étale presheaves on an open étale groupoid
1985	Michael Schlessinger–Jim Stasheff	\| L_∞-algebras
1985	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... –Ross Street	\| Braided monoidal categories
1985	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... –Ross Street	\| Joyal–Street coherence theorem for braided monoidal categories
1985	Paul Ghez–Ricardo Lima–John Roberts	\| C*-categories
1986	Joachim Lambek Joachim Lambek Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :... –Phil Scott	\| Influential book: Introduction to higher order categorical logic
1986	Joachim Lambek Joachim Lambek Joachim Lambek is Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his Ph.D. degree in 1950 with Hans Julius Zassenhaus as advisor. He is called Jim by his friends.- Scholarly work :... –Phil Scott	\| Fundamental theorem of topology: The section-functor Γ and the germ-functor Λ establish a dual adjunction between the category of presheaves and the category of bundles (over the same topological space) which restricts to a dual equivalence of categories (or duality) between corresponding full subcategories of sheaves and of étale bundles
1986	Peter Freyd–David Yetter	\| Constructs the (compact braided) monoidal category of tangles
1986	Vladimir Drinfel'd Vladimir Drinfel'd Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the... –Michio Jimbo Michio Jimbo is a Japanese mathematician, currently a professor at the University of Tokyo. He is a grandson of the linguist Kaku Jimbo.After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University...	\| Quantum groups: In other words quasitriangular Hopf algebra Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally... s. The point is that the categories of representations of quantum groups are tensor categories Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... with extra structure. They are used in construction of quantum invariant Quantum invariant In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.-List of invariants:Finite type invariantKontsevich invariantKashaev's invariant... s of knots and links and low dimensional manifolds, representation theory, q-deformation theory, CFT CFT The three-letter abbreviation CFT may refer to:-2β-Carbomethoxy-3β-tropaneCalifornia Federation of TeachersCardholder Funds TransferCefatrizineChichester Festival TheatreClass field theoryClassical field theory... , integrable systems. The invariants are constructed from braided monoidal categories that are categories of representations of quantum groups. The underlying structure of a TQFT is a modular category of representations of a quantum group
1986	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...	\| Mathematics, form and function Mathematics, Form and Function Mathematics, Form and Function is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.- Mac Lane's relevance to the philosophy of mathematics :... (a foundation of mathematics)
1987	Jean-Yves Girard Jean-Yves Girard Jean-Yves Girard is a French logician working in proof theory. His contributions include a proof of strong normalization in a system of second-order logic called system F; the invention of linear logic; the geometry of interaction; and ludics...	\| Linear logic Linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter... : The internal logic of a linear category (an enriched category Enriched category In category theory and its applications to mathematics, enriched category is a generalization of category that abstracts the set of morphisms associated with every pair of objects to an opaque object in some fixed monoidal category of "hom-objects" and then defines composition and identity solely... with its Hom-sets being linear spaces)
1987	Peter Freyd	\| Freyd representation theorem for Grothendieck toposes
1987	Ross Street	\| Definition of the nerve of a weak n-category and thus obtaining the first definition of weak n-category using simplices
1987	Ross Street–John Roberts	\| Formulates Street–Roberts conjecture: Strict ω-categories are equivalent to complicial sets
1987	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... –Ross Street–Mei Chee Shum	\| Ribbon categories: A balanced rigid braided monoidal category Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
1987	Ross Street	\| n-computads
1987	Iain Aitchison	\| Bottom up Pascal triangle algorithm for computing nonabelian n-cocycle conditions for nonabelian cohomology
1987	Vladimir Drinfel'd Vladimir Drinfel'd Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the... -Gérard Laumon Gérard Laumon Gérard Laumon is a French mathematician. He studied at the École Normale Supérieure and Paris-Sud 11 University, Orsay.In 2004 Laumon and Ngô Bảo Châu received the Clay Research Award for the proof of the Langlands and Shelstad's Fundamental Lemma for unitary groups, a key component in the...	\| Formulates geometric Langlands program Langlands program The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
1987	Vladimir Turaev	\| Starts quantum topology Quantum topology Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of... by using quantum groups and R-matrices R-matrix The term R-matrix has several meanings, depending on the field of study.The term R-matrix is used in connection with the Yang–Baxter equation. This is an equation which was first introduced in the field of statistical mechanics, taking its name from independent work of C. N. Yang and R. J.... to giving an algebraic unification of most of the known knot polynomial Knot polynomial In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W... s. Especially important was Vaughan Jones Vaughan Jones Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize... and Edward Witten Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study.... s work on the Jones polynomial
1988	Alex Heller	\| Heller axioms for homotopy theory as a special abstract hyperfunctor. A feature of this approach is a very general localization Localization of a category In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before...
1988	Alex Heller	\| Heller derivator Derivator In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories and provide at the same time a language for homotopical algebra.... s, the dual of Grothendieck derivator Derivator In mathematics, derivators are a proposed new framework for homological algebra and various generalisations. It is intended to address the perceived deficiencies of derived categories and provide at the same time a language for homotopical algebra.... s
1988	Alex Heller	\| Gives a global closed model structure Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... on the category of simplicial presheaves. John Jardine has also given a model structure for the category of simplicial presheaves
1988	Graeme Segal Graeme Segal Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...	\| Elliptic objects: A functor that is a categorified version of a vector bundle equipped with a connection, it is a 2D parallel transport for strings
1988	Graeme Segal Graeme Segal Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil...	\| Conformal field theory CFT Conformal field theory A conformal field theory is a quantum field theory that is invariant under conformal transformations... : A symmetric monoidal functor Z:nCob_C→Hilb satisfying some axioms
1988	Edward Witten Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....	\| Topological quantum field theory TQFT: A monoidal functor Z:nCob→Hilb satisfying some axioms
1988	Edward Witten Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....	\| Topological string theory Topological string theory In theoretical physics, topological string theory is a simplified version of string theory. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount of supersymmetry...
1989	Hans Baues	\| Influential book: Algebraic homotopy
1989	Michael Makkai Michael Makkai Michael Makkai is a Canadian mathematician, specializing in mathematical logic. He works in model theory, category theory, algebraic logic, and in the theory of topoi. He graduated from the Eötvös Loránd University, Budapest, then worked at the Mathematical Institute of the Hungarian Academy of... -Robert Paré	\| Accessible categories Accessible category The theory of accessible categories was introduced in 1989 by mathematicians Michael Makkai and Robert Paré in the setting of category theory. While the original motivation came from model theory, a branch of mathematical logic, it turned out that accessible categories have applications in homotopy... : Categories with a "good" set of generators Generator (category theory) In category theory in mathematics a generator of a category \mathcal C is an object G of the category, such that for any two different morphisms f, g: X \rightarrow Y in \mathcal C, there is a morphism h : G \rightarrow X, such that the compositions f \circ h \neq g \circ h.Generators are central... allowing to manipulate large categories as if they were small categories, without the fear of encountering any set-theoretic paradoxes. Locally presentable categories are complete accessible categories. Accessible categories are the categories of models of sketches. The name comes from that these categories are accessible as models of sketches.
1989	Edward Witten Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....	\| Witten functional integral formalism and Witten invariants Quantum invariant In the mathematical field of knot theory, a quantum invariant of a knot or link is a linear sum of colored Jones polynomial of surgery presentations of the knot complement.-List of invariants:Finite type invariantKontsevich invariant*Kashaev's invariant... for manifolds.
1990	Peter Freyd	\| Allegories (category theory) Allegory (category theory) In mathematical category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation... : An abstraction of the category of sets and relations as morphisms, it bears the same resemblance to binary relations as categories do to functions and sets. It is a category in which one has in addition to composition a unary operation reciprocation R° and a partial binary operation intersection R ∩ S, like in the category of sets with relations as morphisms (instead of functions) for which a number of axioms are required. It generalizes the relation algebra Relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation... to relations between different sorts.
1990	Nicolai Reshetikhin Nicolai Reshetikhin Nicolai Yuryevich Reshetikhin is a mathematical physicist, currently a professor of mathematics at the University of California, Berkeley and a professor of mathematical physics at the University of Amsterdam. His research is in the fields of low-dimensional topology, representation theory, and... –Vladimir Turaev–Edward Witten Edward Witten Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....	\| Reshetikhin–Turaev–Witten invariants of knots from modular tensor categories of representations of quantum group Quantum group In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra... s.

1991–2000

Year	Contributors	Event
1991	Jean-Yves Girard Jean-Yves Girard Jean-Yves Girard is a French logician working in proof theory. His contributions include a proof of strong normalization in a system of second-order logic called system F; the invention of linear logic; the geometry of interaction; and ludics...	\| Polarization of linear logic Linear logic Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter... .
1991	Ross Street	\| Parity complexes. A parity complex generates a free ω-category.
1991	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... -Ross Street	\| Formalization of Penrose string diagram String diagram In category theory, string diagrams are a way of representing 2-cells in 2-categories. The idea is to represent structures of dimension d by structures of dimension 2-d, using the Poincaré duality... s to calculate with abstract tensors in various monoidal categories Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... with extra structure. The calculus now depends on the connection with low dimensional topology.
1991	Ross Street	\| Definition of the descent strict ω-category of a cosimplicial strict ω-category.
1991	Ross Street	\| Top down excision of extremals algorithm for computing nonabelian n-cocycle conditions for nonabelian cohomology.
1992	Yves Diers	\| Axiomatic categorical geometry using algebraic-geometric categories and algebraic-geometric functors.
1992	Saunders Mac Lane Saunders Mac Lane Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:... -Ieke Moerdijk Ieke Moerdijk Izak Moerdijk is professor of Mathematics at the Mathematisch Instituut of the Radboud University Nijmegen. He is the author of several influential books.-Selected works:...	\| Influential book: Sheaves in geometry and logic.
1992	John Greenlees-Peter May J. Peter May Jon Peter May is an American mathematician, working in the fields of algebraic topology, category theory, homotopy theory, and the foundational aspects of spectra. He is known, in particular, for inventing the term operads and the May spectral sequence.He received a B.A. from Swarthmore College in...	\| Greenlees-May duality
1992	Vladimir Turaev	\| Modular tensor categories. Special tensor categories Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... that arise in constructing knot invariant Knot invariant In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the... s, in constructing TQFT Topological quantum field theory A topological quantum field theory is a quantum field theory which computes topological invariants.... s and CFT Conformal field theory A conformal field theory is a quantum field theory that is invariant under conformal transformations... s, as truncation (semisimple quotient) of the category of representations of a quantum group Quantum group In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra... (at roots of unity), as categories of representations of weak Hopf algebra Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally... s, as category of representations of a RCFT Conformal field theory A conformal field theory is a quantum field theory that is invariant under conformal transformations... .
1992	Vladimir Turaev-Oleg Viro Oleg Viro Oleg Viro is a mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory....	\| Turaev-Viro state sum models based on spherical categories (the first state sum models) and Turaev-Viro state sum invariants for 3-manifolds.
1992	Vladimir Turaev	\| Shadow world of links: Shadows of links give shadow invariants of links by shadow state sums.
1993	Ruth Lawrence Ruth Lawrence Ruth Elke Lawrence-Naimark is an Associate Professor of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, and a researcher in knot theory and algebraic topology. Outside academia, she is best known for being a child prodigy in mathematics.- Youth :Ruth Lawrence...	\| Extended TQFTs
1993	David Yetter-Louis Crane	\| Crane-Yetter state sum models based on ribbon categories and Crane-Yetter state sum invariants for 4-manifolds.
1993	Kenji Fukaya	\| A_∞-categories and A_∞-functors: Most commonly in homological algebra Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and... , a category with several compositions such that the first composition is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. (associative up to higher homotopy). A stands for associative. Def: A category C such that 1) for all X,Y in Ob(C) the Hom-sets Hom_C(X,Y) are finite dimensional chain complex Chain complex In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or... es of Z-graded modules 2) for all objects X₁,...,X_n in Ob(C) there is a family of linear composition maps (the higher compositions) m_n : Hom_C(X₀,X₁) ⊗ Hom_C(X₁,X₂) ⊗ ... ⊗ Hom_C(X_n-1,X_n) → Hom_C(X₀,X_n) of degree n-2 (homological grading convention is used) for n≥1 3) m₁ is the differential on the chain complex Hom_C(X,Y) 4) m_n satisfy the quadratic A_∞-associativity equation for all n≥0. m₁ and m₂ will be chain maps but the compositions m_i of higher order are not chain maps; nevertheless they are Massey product Massey product In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product.-Massey triple product:... s. In particular it is a linear category. Examples are the Fukaya category Fuk(X) and loop space ΩX where X is a topological space and A_∞-algebras as A_∞-categories with one object. When there are no higher maps (trivial homotopies) C is a dg-category. Every A_∞-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology. The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of A_∞-categories and A_∞-functors. Many features of A_∞-categories and A_∞-functors come from the fact that they form a symmetric closed multicategory Multicategory In mathematics , a multicategory is a generalization of the concept of category that allows morphisms of multiple arity... , which is revealed in the language of comonads Monad (category theory) In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations... . From a higher dimensional perspective A_∞-categories are weak ω-categories with all morphisms invertible. A_∞-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.
1993	John Barret-Bruce Westbury	\| Spherical categories: Monoidal categories Monoidal category In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism... with duals for diagrams on spheres instead for in the plane.
1993	Maxim Kontsevich Maxim Kontsevich Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...	\| Kontsevich invariant Kontsevich invariant In the mathematical theory of knots, the Kontsevich invariant, also known as the Kontsevich integral, of an oriented framed link is the universal finite type invariant in the sense that any coefficient of the Kontsevich invariant is a finite type invariant, and any finite type invariant can be... s for knots (are perturbation expansion Feynman integrals for the Witten functional integral) defined by the Kontsevich integral. They are the universal Vassiliev invariants for knots.
1993	Daniel Freed	\| A new view on TQFT using modular tensor categories that unifies three approaches to TQFT (modular tensor categories from path integrals).
1994	Francis Borceux	\| Handbook of Categorical Algebra (3 volumes).
1994	Jean Bénabou-Bruno Loiseau	\| Orbitals in a topos.
1994	Maxim Kontsevich Maxim Kontsevich Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...	\| Formulates the homological mirror symmetry Homological mirror symmetry Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.-History:... conjecture: X a compact symplectic manifold with first Chern class Chern class In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:... c₁(X)=0 and Y a compact Calabi–Yau manifold are mirror pairs if and only if D(Fuk_X) (the derived category of the Fukaya triangulated category of X concocted out of Lagrangian cycles with local systems) is equivalent to a subcategory of D^b(Coh_Y) (the bounded derived category of coherent sheaves on Y).
1994	Louis Crane-Igor Frenkel Igor Frenkel Igor Borisovich Frenkel is a mathematician at Yale University working in representation theory and mathematical physics.-Biography:...	\| Hopf categories and construction of 4D TQFTs by them.
1994	John Fischer	\| Defines the 2-category 2-category In category theory, a 2-category is a category with "morphisms between morphisms"; that is, where each hom set itself carries the structure of a category... of 2-knots (knotted surfaces).
1995	Bob Gordon-John Power-Ross Street	\| Tricategories Tricategory In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory.Whereas a weak 2-category is said to be a bicategory [Benabou 1967], a weak 3-category is said to be a tricategory .Tetracategories are the corresponding notion in dimension four... and a corresponding coherence theorem Coherence theorem In mathematics and particularly category theory, a coherence theorem is a tool for proving a coherence condition. Typically a coherence condition requires an infinite number of equalities among compositions of structure maps... : Every weak 3-category is equivalent to a Gray 3-category.
1995	Ross Street-Dominic Verity	\| Surface diagrams for tricategories.
1995	Louis Crane	\| Coins categorification Categorification In mathematics, categorification refers to the process of replacing set-theoretic theorems by category-theoretic analogues. Categorification, when done successfully, replaces sets by categories, functions with functors, and equations by natural isomorphisms of functors satisfying additional... leading to the categorical ladder.
1995	Sjoerd Crans	\| A general procedure of transferring closed model structure Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... s on a category along adjoint functor pairs to another category.
1995	André Joyal André Joyal André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. Joyal was born in Drummondville . He has three children and lives in Montreal.- Main research :... -Ieke Moerdijk Ieke Moerdijk Izak Moerdijk is professor of Mathematics at the Mathematisch Instituut of the Radboud University Nijmegen. He is the author of several influential books.-Selected works:...	\| AST Algebraic set theory: Also sometimes called categorical set theory. It was developed from 1988 by André Joyal and Ieke Moerdijk, and was first presented in detail as a book in 1995 by them. AST is a framework based on category theory to study and organize set theories and to construct models of set theories. The aim of AST is to provide a uniform categorical semantics or description of set theories of different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded,...), the various constructions of the cumulative hierarchy of sets, forcing models, sheaf models and realisability models. Instead of focusing on categories of sets AST focuses on categories of classes. The basic tool of AST is the notion of a category with class structure (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a universal object (a universe Universe (mathematics) In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation... )) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of ZF-algebras (Zermelo-Fraenkel algebra) and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first order logic of elementary set theory on the other. The subcategory of sets in a class category is an elementary topos and every elementary topos occurs as sets in a class category. The class category itself always embeds into the ideal completion of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory BIST that is logically complete with respect to class category models. Therefore class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The ZF-axioms Zermelo–Fraenkel set theory In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes... are nothing but a description of the free ZF-algebra just as the Peano axioms are a description of the free monoid on one generator. In this perspective the models of set theory are algebras for a suitably presented algebraic theory Theory (mathematical logic) In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom... and many familiar set theoretic conditions (such as well foundedness) are related to familiar algebraic conditions (such as freeness). Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes.
1995	Michael Makkai Michael Makkai Michael Makkai is a Canadian mathematician, specializing in mathematical logic. He works in model theory, category theory, algebraic logic, and in the theory of topoi. He graduated from the Eötvös Loránd University, Budapest, then worked at the Mathematical Institute of the Hungarian Academy of...	\| SFAM Structuralist foundation of abstract mathematics. In SFAM the universe consists of higher dimensional categories, functors are replaced by saturated anafunctors, sets are abstract sets, the formal logic for entities is FOLDS (first-order logic with dependent sorts) in which the identity relation is not given a priori by first order axioms but derived from within a context.
1995	John Baez-James Dolan	\| Opetopic sets (opetopes) based on operads. Weak n-categories Weak n-category In category theory, weak n-categories are a generalization of the notion of n-category where composition is not strictly associative but only associative up to coherent equivalence. There is currently much work to determine what the coherence laws should be for those. Weak n-categories have become... are n-opetopic sets.
1995	John Baez-James Dolan	\| Introduced the periodic table of mathematics which identifies k-tuply monoidal n-categories. It mirrors the table of homotopy groups of the spheres.
1995	John Baez-James Dolan	\| Outlined a program in which n-dimensional TQFTs are described as n-category representations.
1995	John Baez-James Dolan	\| Proposed n-dimensional deformation quantization.
1995	John Baez-James Dolan	\| Tangle hypothesis: The n-category of framed n-tangles in n + k dimensions is (n+k)-equivalent to the free weak k-tuply monoidal n-category with duals on one object.
1995	John Baez-James Dolan	\| Cobordism hypothesis (Extended TQFT hypothesis I): The n-category of which n-dimensional extended TQFTs are representations, nCob, is the free stable weak n-category with duals on one object.
1995	John Baez-James Dolan	\| Stabilization hypothesis: After suspending a weak n-category n + 2 times, further suspensions have no essential effect. The suspension functor S:nCat_k→nCat_k+1 is an equivalence of categories for k = n + 2.
1995	John Baez-James Dolan	\| Extended TQFT hypothesis II: An n-dimensional unitary extended TQFT is a weak n-functor, preserving all levels of duality, from the free stable weak n-category with duals on one object to nHilb.
1995	Valentin Lychagin	\| Categorical quantization
1995	Pierre Deligne Pierre Deligne - See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :... -Vladimir Drinfel'd Vladimir Drinfel'd Vladimir Gershonovich Drinfel'd is a Ukrainian and Soviet mathematician at the University of Chicago.The work of Drinfeld related algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the... -Maxim Kontsevich Maxim Kontsevich Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...	\| Derived algebraic geometry with derived schemes and derived moduli stacks. A program of doing algebraic geometry and especially moduli problems in the derived category Derived category In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C... of schemes or algebraic varieties instead of in their normal categories.
1997	Maxim Kontsevich Maxim Kontsevich Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...	\| Formal deformation quantization theorem: Every Poisson manifold Poisson manifold In mathematics, a Poisson manifold is a differentiable manifold M such that the algebra C^\infty\, of smooth functions over M is equipped with a bilinear map called the Poisson bracket, turning it into a Poisson algebra... admits a differentiable star product and they are classified up to equivalence by formal deformations of the Poisson structure.
1998	Claudio Hermida-Michael-Makkai-John Power	\| Multitopes, Multitopic sets.
1998	Carlos Simpson	\| Simpson conjecture: Every weak ∞-category is equivalent to a ∞-category in which composition and exchange laws are strict and only the unit laws are allowed to hold weakly. It is proven for 1,2,3-categories with a single object.
1998	André Hirschowitz-Carlos Simpson	\| Give a model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... structure on the category of Segal categories. Segal categories Segal category In mathematics, a Segal category is a model of an infinity category introduced by , based on work of Graeme Segal in 1974.... are the fibrant-cofibrant objects and Segal maps are the weak equivalences Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... . In fact they generalize the definition to that of a Segal n-category and give a model structure for Segal n-categories for any n ≥ 1.
1998	Chris Isham-Jeremy Butterfield	\| Kochen-Specker theorem Kochen-Specker theorem In quantum mechanics, the Kochen–Specker theorem is a "no go" theorem proved by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden variable theories which try to explain the apparent randomness of quantum mechanics as a deterministic model... in topos theory of presheaves: The spectral presheaf (the presheaf that assigns to each operator its spectrum) has no global element Global element In category theory, a global element of an object A from a category is a morphismwhere 1 is a terminal object of the category. Roughly speaking, global elements are a generalization of the notion of “elements” from the category of sets, and they can be used to import set-theoretic... s (global sections) but may have partial elements or local elements. A global element is the analogue for presheaves of the ordinary idea of an element of a set. This is equivalent in quantum theory to the spectrum of the C*-algebra of observables in a topos having no points.
1998	Richard Thomas	\| Richard Thomas, a student of Simon Donaldson Simon Donaldson Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London... , introduces Donaldson–Thomas invariants which are systems of numerical invariants of complex oriented 3-manifolds X, analogous to Donaldson invariants in the theory of 4-manifolds. They are certain weighted Euler characteristics of the moduli space of sheaves on X and "count" Gieseker semistable coherent sheaves Coherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with... with fixed Chern character on X. Ideally the moduli spaces should be a critical sets of holomorphic Chern–Simons functions and the Donaldson–Thomas invariants should be the number of critical points of this function, counted correctly. Currently such holomorphic Chern–Simons functions exist at best locally.
1998	John Baez	\| Spin foam models Spin foam In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral description of quantum gravity... : A 2-dimensional cell complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams are functors between spin network categories. Any slice of a spin foam gives a spin network.
1998	John Baez–James Dolan	\| Microcosm principle: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure.
1998	Alexander Rosenberg	\| Noncommutative schemes: The pair (Spec(A),O_A) where A is an abelian category Abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative... and to it is associated a topological space Spec(A) together with a sheaf of rings O_A on it. In the case when A = QCoh(X) for X a scheme the pair (Spec(A),O_A) is naturally isomorphic to the scheme (X^Zar,O_X) using the equivalence of categories QCoh(Spec(R))=Mod_R. More generally abelian categories or triangulated categories or dg-categories or A_∞-categories should be regarded as categories of quasicoherent sheaves (or complexes of sheaves) on noncommutative schemes. This is a starting point in noncommutative algebraic geometry Noncommutative algebraic geometry Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them... . It means that one can think of the category A itself as a space. Since A is abelian it allows to naturally do homological algebra Homological algebra Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and... on noncommutative schemes and hence sheaf cohomology Sheaf cohomology In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F... .
1998	Maxim Kontsevich Maxim Kontsevich Maxim Lvovich Kontsevich is a Russian mathematician. He is a professor at the Institut des Hautes Études Scientifiques and a distinguished professor at the University of Miami...	\| Calabi–Yau categories: A linear category with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map. If X is a smooth projective Calabi—Yau variety of dimension d then D^b(Coh(X)) is a unital Calabi–Yau A_∞-category of Calabi–Yau dimension d. A Calabi–Yau category with one object is a Frobenius algebra Frobenius algebra In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in... .
1999	Joseph Bernstein Joseph Bernstein Joseph Bernstein is an Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory.... –Igor Frenkel Igor Frenkel Igor Borisovich Frenkel is a mathematician at Yale University working in representation theory and mathematical physics.-Biography:... –Mikhail Khovanov Mikhail Khovanov Mikhail Khovanov is a professor of mathematics at Columbia University. He earned a PhD in mathematics from Yale University in 1997, where he studied under Igor Frenkel. His interests include knot theory and algebraic topology...	\| Temperley–Lieb categories: Objects are enumerated by nonnegative integers. The set of homomorphisms from object n to object m is a free R-module with a basis over a ring R. R is given by the isotopy classes of systems of (\|n\| + \|m\|)/2 simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs \|n\| points on the bottom and \|m\| points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley–Lieb categories are categorized Temperley–Lieb algebras.
1999	Moira Chas–Dennis Sullivan Dennis Sullivan Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...	\| Constructs String topology String topology String topology, a branch of mathematics, is the study of algebraic structures on the homology of free loop spaces. The field was started by Chas and Sullivan in 1999 .-Motivation:... by cohomology. This is string theory on general topological manifolds.
1999	Mikhail Khovanov Mikhail Khovanov Mikhail Khovanov is a professor of mathematics at Columbia University. He earned a PhD in mathematics from Yale University in 1997, where he studied under Igor Frenkel. His interests include knot theory and algebraic topology...	\| Khovanov homology Khovanov homology In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial.... : A homology theory for knots such that the dimensions of the homology groups are the coefficients of the Jones polynomial of the knot.
1999	Vladimir Turaev	\| Homotopy quantum field theory HQFT
1999	Vladimir Voevodsky Vladimir Voevodsky Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :... –Fabien Morel	\| Constructs the homotopy category of schemes A¹ homotopy theory In algebraic geometry and algebraic topology, a branch of mathematics, A1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky... .
1999	Ronald Brown Ronald Brown (mathematician) Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and journal articles.-Education and career:... –George Janelidze	\| 2-dimensional Galois theory
2000	Vladimir Voevodsky Vladimir Voevodsky Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...	\| Gives two constructions of motivic cohomology Motivic cohomology Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry... of varieties, by model categories in homotopy theory and by a triangulated category of DM-motives.
2000	Yasha Eliashberg–Alexander Givental Alexander Givental Alexander Givental is a Russian American mathematician working in the area of symplectic topology, singularity theory and their relations to topological string theories. He got his Ph. D. under the supervision of V. I. Arnold. He, first, proved the mirror conjecture for toric Calabi-Yau manifolds,... –Helmut Hofer	\| Symplectic field theory SFT: A functor Z from a geometric category of framed Hamiltonian structures and framed cobordisms between them to an algebraic category of certain differential D-modules and Fourier integral operators between them and satisfying some axioms.
2000	Paul Taylor	\| ASD (Abstract Stone duality): A reaxiomatisation of the space and maps in general topology in terms of λ-calculus Lambda calculus In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus... of computable continuous functions and predicates that is both constructive and computable. The topology on a space is treated not as a lattice, but as an exponential object Exponential object In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories... of the same category as the original space, with an associated λ-calculus. Every expression in the λ-calculus denotes both a continuous function and a program. ASD does not use the category of sets Category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B... , but the full subcategory of overt discrete objects plays this role (an overt object is the dual to a compact object), forming an arithmetic universe (pretopos with lists) with general recursion.

2001–present

Year	Contributors	Event
2001	Charles Rezk	\| Constructs a model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... with certain generalized Segal categories Segal category In mathematics, a Segal category is a model of an infinity category introduced by , based on work of Graeme Segal in 1974.... as the fibrant objects, thus obtaining a model for a homotopy theory of homotopy theories. Complete Segal spaces are introduced at the same time.
2001	Charles Rezk	\| Model toposes and their generalization homotopy toposes (a model topos without the t-completness assumption).
2002	Bertrand Toën-Gabriele Vezzosi	\| Segal toposes coming from Segal topologies, Segal sites and stacks over them.
2002	Bertrand Toën-Gabriele Vezzosi	\| Homotopical algebraic geometry: The main idea is to extend schemes Scheme (mathematics) In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern... by formally replacing the rings with any kind of "homotopy-ring-like object". More precisely this object is a commutative monoid in a symmetric monoidal category endowed with a notion of equivalences which are understood as "up-to-homotopy monoid" (e.g. E_∞-rings).
2002	Peter Johnstone Peter Johnstone (mathematician) Peter Tennant Johnstone is Professor of the Foundations of Mathematics at St. John's College, Cambridge. He invented or developed a broad range of fundamental ideas in topos theory.-Books by Peter Johnstone:. –-External links:...	\| Influential book: sketches of an elephant - a topos theory compendium. It serves as an encyclopedia of topos Topos In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space... theory (2/3 volumes published as of 2008).
2002	Dennis Gaitsgory Dennis Gaitsgory Dennis Gaitsgory is a mathematician at Harvard University known for his research on the geometric Langlands program. Born in what is now Moldova, he grew up in Tajikistan, before studying at Tel Aviv University under Joseph Bernstein. He received his doctorate in 1997 for a thesis entitled... -Kari Vilonen-Edward Frenkel	\| Proves the geometric Langlands program Langlands program The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by .... for GL(n) over finite fields.
2003	Denis-Charles Cisinski	\| Makes further work on ABC model categories and brings them back into light. From then they are called ABC model categories after their contributors.
2004	Dennis Gaitsgory Dennis Gaitsgory Dennis Gaitsgory is a mathematician at Harvard University known for his research on the geometric Langlands program. Born in what is now Moldova, he grew up in Tajikistan, before studying at Tel Aviv University under Joseph Bernstein. He received his doctorate in 1997 for a thesis entitled...	\| Extended the proof of the geometric Langlands program Langlands program The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by .... to include GL(n) over C. This allows to consider curves over C instead of over finite fields in the geometric Langlands program.
2004	Mario Caccamo	\| Formal category theoretical expanded λ-calculus for categories.
2004	Francis Borceux-Dominique Bourn	\| Homological categories
2004	William Dwyer-Philips Hirschhorn-Daniel Kan Daniel Kan Daniel Marinus Kan is a mathematician working in homotopy theory. He has been a prolific contributor to the field for the last five decades, having authored or coauthored several dozen research papers and monographs. The general theme of his career has been abstract homotopy theory.He is an... -Jeffrey Smith	\| Introduces in the book: Homotopy limit functors on model categories and homotopical categories, a formalism of homotopical categories and homotopical functors (weak equivalence preserving functors) that generalize the model category Model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes... formalism of Daniel Quillen. A homotopical category has only a distinguished class of morphisms (containing all isomorphisms) called weak equivalences and satisfy the two out of six axiom. This allow to define homotopical versions of initial and terminal objects, limit Limit (category theory) In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits.... and colimit functors (that are computed by local constructions in the book), completeness Complete category In mathematics, a complete category is a category in which all small limits exist. That is, a category C is complete if every diagram F : J → C where J is small has a limit in C. Dually, a cocomplete category is one in which all small colimits exist... and cocompleteness, adjunction Adjoint functors In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency... s, Kan extension Kan extension Kan extensions are universal constructs in category theory, a branch of mathematics. They are closely related to adjoints, but are also related to limits and ends. They are named after Daniel M... s and universal properties Universal property In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment... .
2004	Dominic Verity	\| Proves the Street-Roberts conjecture.
2004	Ross Street	\| Definition of the descent weak ω-category of a cosimplicial weak ω-category.
2004	Ross Street	\| Characterization theorem for cosmoses: A bicategory M is a cosmos iff there exists a base bicategory W such that M is biequivalent to Mod_W. W can be taken to be any full subbicategory of M whose objects form a small Cauchy generator Generator (category theory) In category theory in mathematics a generator of a category \mathcal C is an object G of the category, such that for any two different morphisms f, g: X \rightarrow Y in \mathcal C, there is a morphism h : G \rightarrow X, such that the compositions f \circ h \neq g \circ h.Generators are central... .
2004	Ross Street-Brian Day	\| Quantum categories and quantum groupoids: A quantum category over a braided monoidal category V is an object R with an opmorphism h:R^op ⊗ R → A into a pseudomonoid A such that h^* is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, h and A lie in the autonomous monoidal bicategory Comod(V)^co of comonoids. Comod(V)=Mod(V^op)^coop. Quantum categories were introduced to generalize Hopf algebroids and groupoids. A quantum groupoid is a Hopf algebra Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally... with several objects.
2004	Stephan Stolz-Peter Teichner	\| Definition of nD QFT Quantum field theory Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and... of degree p parametrized by a manifold.
2004	Stephan Stolz-Peter Teichner	\| Graeme Segal Graeme Segal Graeme Bryce Segal is a British mathematician, and professor at the University of Oxford.Segal was educated at the University of Sydney, where he received his BSc degree in 1961. He went on to receive his D.Phil... proposed in the 1980s to provide a geometric construction of elliptic cohomology Elliptic cohomology In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.-History and motivation:Historically, elliptic cohomology arose from the study of elliptic genera... (the precursor to tmf Topological modular forms In mathematics, the spectrum of topological modular forms describes a generalized cohomology theory whose coefficient ring is related to the graded ring of holomorphic modular forms with integral cusp expansions... ) as some kind of moduli space of CFTs. Stephan Stolz and Peter Teichner continued and expanded these ideas in a program to construct TMF Topological modular forms In mathematics, the spectrum of topological modular forms describes a generalized cohomology theory whose coefficient ring is related to the graded ring of holomorphic modular forms with integral cusp expansions... as a moduli space of supersymmetric Euclidean field theories. They conjectured a Stolz-Teichner picture (analogy) between classifying space Classifying space In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G... s of cohomology theories in the chromatic filtration (de Rham cohomology,K-theory,Morava K-theories) and moduli spaces of supersymmetric QFTs parametrized by a manifold (proved in 0D and 1D).
2005	Peter Selinger	\| Dagger categories Dagger category In mathematics, a dagger category is a category equipped with a certain structure called dagger or involution... and dagger functors. Dagger categories seem to be part of a larger framework involving n-categories with duals.
2005	Peter Ozsváth Peter Ozsváth Peter Steven Ozsváth is a professor of mathematics at Princeton University. He created, along with Zoltán Szabó, Heegaard Floer homology, a homology theory for 3-manifolds.... -Zoltán Szabó	\| Knot Floer homology
2006	P. Carrasco-A.R. Garzon-E.M. Vitale	\| Categorical crossed modules
2006	Aslak Bakke Buan–Robert Marsh–Markus Reineke–Idun Reiten Idun Reiten Idun Reiten is a Norwegian professor of mathematics. She is considered to be one of Norway's greatest mathematicians today.-Career:She took her PhD degree at the University of Illinois in 1971... –Gordana Todorov	\| Cluster categories: Cluster categories are a special case of triangulated Calabi–Yau categories of Calabi–Yau dimension 2 and a generalization of cluster algebra Cluster algebra Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.... s.
2006	Jacob Lurie Jacob Lurie Jacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...	\| Monumental book: Higher topos theory: In its 940 pages Jacob Lurie generalize the common concepts of category theory to higher categories and defines n-toposes, ∞-toposes, sheaves of n-types, ∞-sites, ∞-Yoneda lemma Yoneda lemma In mathematics, specifically in category theory, the Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory... and proves Lurie characterization theorem for higher dimensional toposes. Luries theory of higher toposes can be interpreted as giving a good theory of sheaves taking values in ∞-categories. Roughly an ∞-topos is an ∞-category which looks like the ∞-category of all homotopy types. In a topos mathematics can be done. In a higher topos not only mathematics can be done but also "n-geometry", which is higher homotopy theory. The topos hypothesis is that the (n+1)-category nCat is a Grothendieck (n+1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting.
2006	Marni Dee Sheppeard	\| Quantum toposes
2007	Bernhard Keller-Thomas Hugh	\| d-cluster categories
2007	Dennis Gaitsgory Dennis Gaitsgory Dennis Gaitsgory is a mathematician at Harvard University known for his research on the geometric Langlands program. Born in what is now Moldova, he grew up in Tajikistan, before studying at Tel Aviv University under Joseph Bernstein. He received his doctorate in 1997 for a thesis entitled... -Jacob Lurie Jacob Lurie Jacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...	\| Presents a derived version of the geometric Satake equivalence and formulates a geometric Langlands duality for quantum group Quantum group In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra... s. The geometric Satake equivalence realized the category of representations of the Langlands dual group ^LG in terms of spherical perverse sheaves (or D-module D-module In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations... s) on the affine Grassmannian Grassmannian In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological... Gr_G = G((t))/Gt T T is the 20th letter in the basic modern Latin alphabet. It is the most commonly used consonant and the second most common letter in the English language.- History :Taw was the last letter of the Western Semitic and Hebrew alphabets... of the original group G.
2008	Ieke Moerdijk Ieke Moerdijk Izak Moerdijk is professor of Mathematics at the Mathematisch Instituut of the Radboud University Nijmegen. He is the author of several influential books.-Selected works:... -Clemens Berger	\| Extends and improved the definition of Reedy category to become invariant under equivalence of categories Equivalence of categories In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics... .
2008	Michael J. Hopkins Michael J. Hopkins Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.-Life:He received his Ph.D. from Northwestern University in 1984 under the direction of Mark Mahowald. In 1984 he also received his D.Phil... –Jacob Lurie Jacob Lurie Jacob Alexander Lurie is an American mathematician, who is currently a professor at Harvard University.-Life:While in school, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994...	\| Sketch of proof of Baez-Dolan tangle hypothesis and Baez-Dolan cobordism hypothesis which classify extended TQFT in all dimensions.

Timeline to 1945: before the definitions

1945–1970

1971–1980

1981–1990

1991–2000

2001–present

See also