In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
topos (plural "topoi" or "toposes") is a type of
categoryIn mathematics, a category is an algebraic structure consisting of a collection of "objects", linked together by a collection of "arrows" that have two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Objects and arrows may...
that behaves like the category of sheaves of sets on a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
. For a discussion of the history of topos theory, see the article
Background and genesis of topos theoryThis page gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given...
.
Grothendieck topoi (topoi in geometry)
Since the introduction of sheaves into mathematics in the 1940s a major theme has been to study a space by studying sheaves on that space. This idea was expounded by
Alexander GrothendieckAlexander Grothendieck is considered one of the greatest mathematicians of the 20th century.He is most famous for his revolutionary advances in algebraic geometry, but he has also made major contributions to algebraic topology, number theory, category theory, Galois theory, descent theory,...
by introducing the notion of a
topos. The main utility of this notion is in the abundance of situations in mathematics where topological intuition is very effective but an honest topological space is lacking; it is sometimes possible to find a topos formalizing the intuition. The greatest single success of this programmatic idea to date has been the introduction of the étale topos of a
schemeIn 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...
.
Equivalent formulations
Let
C be a category. A theorem of
GiraudThere is another famous Jean Giraud, comics author.Jean Giraud is a French mathematician.He was a graduate student of Alexander Grothendieck and his book "Cohomologie non abélienne" has become a classic, registering so far 104 citations.From 1993 to 1994, he was deputy director for research of...
states that the following are equivalent:
- There is a small category D and an inclusion C Presh(D) that admits a finite-limit-preserving left adjoint.
- C is the category of sheaves on a Grothendieck site.
- C satisfies Giraud's axioms, below.
A category with these properties is called a "(Grothendieck) topos". Here Presh(
D) denotes the category of contravariant functors from
D to the category of sets; such a contravariant functor is frequently called a
presheafIn category theory, a branch of mathematics, a -valued presheaf on a category is a functor . Often presheaf is defined to be a Set-valued presheaf. If is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological...
.
Giraud's axioms
Giraud's axioms for a category
C are:
- C has a small set of generator
In category theory in mathematics a generator of a category is an object G of the category, such that for any two different morphisms in , there is a morphism , such that the compositions ....
s, and admits all small colimits. Furthermore, colimits commute with fiber products.
- Sums in C are disjoint. In other words, the fiber product of X and Y over their sum is the initial object in C.
- All equivalence relations in C are effective.
The last axiom needs the most explanation. If
X is an object of
C, an equivalence relation
R on
X is a map
R→
X×
X in
C
such that all the maps Hom(
Y,
R)→Hom(
Y,
X)×Hom(
Y,
X) are equivalence relations of sets. Since
C has colimits we may form the coequalizer of the two maps
R→
X; call this
X/
R. The equivalence relation is effective if the canonical map
is an isomorphism.
Examples
Giraud's theorem already gives "sheaves on sites" as a complete list of examples. Note, however, that nonequivalent sites often give
rise to equivalent topoi. As indicated in the introduction, sheaves on ordinary topological spaces motivate many of the basic definitions and results of topos theory.
The category of sets is an important special case: it plays the role of a point in topos theory. Indeed, a set may be thought of as a sheaf on a point.
More exotic examples, and the
raison d'être of topos theory, come from algebraic geometry. To a
schemeIn 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...
and even a stack one may associate an étale topos, an fppf topos, a Nisnevich topos...
Counterexamples
Topos theory is, in some sense, a generalization of classical point-set topology. One should therefore expect to see old and new instances of
pathologicalIn mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive.Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological...
behavior. For instance, there is an example due to
Pierre DelignePierre René, Viscount Deligne is a Belgian mathematician. He is known for fundamental work on the Weil conjectures, leading finally to a complete proof in 1973.- Life :...
of a nontrivial topos that has no points (see below).
Geometric morphisms
If
X and
Y are topoi, a
geometric morphism u:
X→
Y is a pair of adjoint functors (
u∗,
u∗) such that
u∗ preserves finite limits. Note that
u∗ automatically preserves colimits by virtue of having a right adjoint.
By Freyd's adjoint functor theorem, to give a geometric morphism
X →
Y is to give a functor
u∗:
Y →
X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales.
If
X and
Y are topological spaces and
u is a continuous map between them, then the pullback and pushforward operations on sheaves yield a geometric morphism between the associated topoi.
Points of topoi
A point of a topos
X is a geometric morphism from the topos of sets to
X.
If
X is an ordinary space and
x is a point of
X, then the functor that takes a sheaf
F to its stalk
Fx has a right adjoint
(the "skyscraper sheaf" functor), so an ordinary point of
X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map
x:
1 →
X.
Essential geometric morphisms
A geometric morphism (
u∗,
u∗) is
essential if
u∗ has a further left adjoint
u!, or equivalently (by the adjoint functor theorem) if
u∗ preserves not only finite but all small limits.
Ringed topoi
A
ringed topos is a pair
(X,R), where
X is a topos and
R is a commutative ring object in
X. Most of the constructions of
ringed spacesIn mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
go through for ringed topoi. The category of
R-module objects in
X is an
abelian categoryIn 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...
with enough injectives. A more useful abelian category is the subcategory of quasi-coherent
R-modules: these are
R-modules that admit a presentation.
Another important class of ringed topoi, besides ringed spaces, are the etale topoi of Deligne-Mumford stacks.
Homotopy theory of topoi
Michael ArtinMichael Artin is an American mathematician and a professor at MIT, known for his contributions to algebraic geometry.Artin was born in Hamburg, Germany, and brought up in Indiana...
and
Barry MazurBarry Charles Mazur is a professor of mathematics at Harvard University.-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate...
associated to any topos a
pro-simplicial setIn mathematics, a pro-simplicial set is an inverse system of simplicial sets.A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups....
. Using this
inverse systemIn mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inverse system is sometimes called a pro-object in C.-The category of inverse systems:...
of simplicial sets one may
sometimes associate to a homotopy invariant in classical topology an
inverse systemIn mathematics, an inverse system in a category C is a functor from a small cofiltered category I to C. An inverse system is sometimes called a pro-object in C.-The category of inverse systems:...
of invariants in topos theory.
The pro-simplicial set associated to the etale topos of a scheme is a
pro-finiteIn mathematics, a pro-simplicial set is an inverse system of simplicial sets.A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups....
simplicial set. Its study is called étale homotopy theory.
Introduction
A traditional axiomatic foundation of mathematics is
set theoryThe modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, in which all mathematical objects are ultimately represented by sets (even
functionsIn mathematics, a function is a relation between a given set of elements and another set of elements , which associates each element in the domain with exactly one element in the codomain...
which map between sets). More recent work in
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
allows this foundation to be generalized using toposes; each topos completely defines its own mathematical framework. The category of sets forms a familiar topos, and working within this topos is equivalent to using traditional set theoretic mathematics. But one could instead choose to work with many alternative toposes. A standard formulation of the
axiom of choiceIn mathematics, the axiom of choice, or AC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins and...
makes sense in any topos, and there are toposes in which it is invalid. Constructivists will be interested to work in a topos without the
law of excluded middleIn logic, The Law of excluded middle, also known as the Principle of excluded middle or Excluded middle is the principle that for any proposition, either that proposition is true, or its negation is. The principle can be expressed in either a logical or a semantical form. The semantical form...
. If symmetry under a particular
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is of importance, one can use the topos consisting of all
G-setsIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
.
It is also possible to encode an
algebraic theoryUniversal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, such as the theory of
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s, as a topos. The individual models of the theory, i.e. the groups in our example, then correspond to
functorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
s from the encoding topos to the category of sets that respect the topos structure.
Formal definition
When used for foundational work a topos will be defined axiomatically; set theory is then treated as a special case of topos theory. Building from category theory, there are multiple equivalent definitions of a topos. The following has the virtue of being concise, if not illuminating:
A topos is a
categoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
which has the following two properties:
- All limits
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....
taken over finite index categories exist.
- Every object has a power object.
From this one can derive that
- All colimits
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....
taken over finite index categories exist.
- The category has a 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...
.
- Any two objects have an 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...
.
- The category is cartesian closed
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...
.
In many applications, the role of the subobject classifier is pivotal, whereas power objects are not. Thus some definitions reverse the roles of what is defined and what is derived.
Explanation
A topos as defined above can be understood as a
cartesian closed categoryIn 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...
for which the notion of subobject of an object has an
elementaryFirst-order logic is a formal logic used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, and predicate logic...
or first-order definition. This notion, as a natural categorical abstraction of the notions of
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of a set,
subgroupIn the mathematical subject known as group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
, and more generally
subalgebraIn algebra , the word "algebra" usually means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of any
algebraic structureIn algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
, predates the notion of topos. It is definable in any category, not just toposes, in
second-orderIn logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory....
language, i.e. in terms of classes of morphisms instead of individual morphisms, as follows. Given two monics
m,
n from respectively
Y and
Z to
X, we say that
m ≤
n when there exists a morphism
p:
Y →
Z for which
np =
m, inducing a
preorderIn mathematics, especially in order theory, preorders are binary relations that satisfy certain conditions. For example, all partial orders and equivalence relations are preorders. The name quasiorder is also common for preorders. Other names are pre-order, quasi-order, and quasi order...
on monics to
X. When
m ≤
n and
n ≤
m we say that
m and
n are equivalent. The subobjects of
X are the resulting equivalence classes of the monics to it.
In a topos "subobject" becomes, at least implicitly, a first-order notion, as follows.
As noted above, a topos is a category
C having all finite limits and hence in particular the empty limit or final object 1. It is then natural to treat morphisms of the form
x: 1 →
X as
elements x ∈
X. Morphisms
f:
X →
Y thus correspond to functions mapping each element
x ∈
X to the element
fx ∈
Y, with application realized by composition.
One might then think to define a subobject of
X as an equivalence class of monics
m:
X' →
X having the same
imageIn mathematics, the image of a subset of a function's domain under the function is the set of all outputs obtained when the function is evaluated at each element of the subset...
or
rangeIn mathematics, the range of a function is the set of all "output" values produced by f.Sometimes it is called the image, or more precisely, the image of the domain of the function. If a function is a surjection then its range is equal to its codomain...
{
mx|
x ∈
X' }. The catch is that two or more morphisms may correspond to the same function, that is, we cannot assume that
C is concrete in the sense that the functor
C(1,-):
C →
Set is faithful. For example the category
Grph of graphs and their associated
homomorphismIn abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s is a topos whose final object 1 is the graph with one vertex and one edge (a self-loop), but is not concrete because the elements 1 →
G of a graph
G correspond only to the self-loops and not the other edges, nor the vertices without self-loops. Whereas the second-order definition makes
G and its set of self-loops (with their vertices) distinct subobjects of
G (unless every edge is, and every vertex has, a self-loop), this image-based one does not. This can be addressed for the graph example and related examples via the
Yoneda LemmaIn 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 . It allows the embedding of any category into a category of functors defined on that...
as described in the Examples section below, but this then ceases to be first-order. Toposes provide a more abstract, general, and first-order solution.
As noted above a topos
C has a
subobject classifierIn 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...
Ω, namely an object of
C with an element
t ∈ Ω, the
generic subobject of
C, having the property that every
monicIn the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....
m:
X' →
X arises as a pullback of the generic subobject along a unique morphism
f:
X → Ω, as per Figure 1. Now the pullback of a monic is a monic, and all elements including
t are monics since there is only one morphism to 1 from any given object, whence the pullback of
t along
f:
X → Ω is a monic. The monics to
X are therefore in bijection with the pullbacks of
t along morphisms from
X to Ω. The latter morphisms partition the monics into equivalence classes each determined by a morphism
f:
X → Ω, the characteristic morphism of that class, which we take to be the subobject of
X characterized or named by
f.
All this applies to any topos, whether or not concrete. In the concrete case, namely
C(1,-) faithful, for example the category of sets, the situation reduces to the familiar behavior of functions. Here the monics
m:
X' →
X are exactly the injections (one-one functions) from
X' to
X, and those with a given image {
mx|
x ∈
X' } constitute the subobject of
X corresponding to the morphism
f:
X → Ω for which
f−1(
t) is that image. The monics of a subobject will in general have many domains, all of which however will be in bijection with each other.
To summarize, this first-order notion of subobject classifier implicitly defines for a topos the same equivalence relation on monics to
X as had previously been defined explicitly by the second-order notion of subobject for any category. The notion of equivalence relation on a class of morphisms is itself intrinsically second-order, which the definition of topos neatly sidesteps by explicitly defining only the notion of subobject
classifier Ω, leaving the notion of subobject of
X as an implicit consequence characterized (and hence namable) by its associated morphism
f:
X → Ω.
Further examples
If
C is a small category, then the
functor categoryIn category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors...
SetC (consisting of all covariant
functorIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms in the category of small categories....
s from
C to sets, with
natural transformationIn category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
s as morphisms) is a topos. For instance, the category
Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos. A graph consists of two sets, an edge set and a vertex set, and two functions
s,t between those sets, assigning to every edge
e its source
s(
e) and target
t(
e).
Grph is thus equivalent to the functor category
SetC, where
C is the category with two objects
E and
V and two morphisms
s,t:
E →
V giving respectively the source and target of each edge.
The categories of finite sets, of finite
G-sets (actions of a group
G on a finite set), and of finite graphs are also toposes.
The
Yoneda LemmaIn 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 . It allows the embedding of any category into a category of functors defined on that...
asserts that
Cop embeds in
SetC as a full subcategory. In the graph example the embedding represents
Cop as the subcategory of
SetC whose two objects are
V' as the one-vertex no-edge graph and
E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from
V' to
E' (both as natural transformations). The natural transformations from
V' to an arbitrary graph (functor)
G constitute the vertices of
G while those from
E' to
G constitute its edges. Although
SetC, which we can identify with
Grph, is not made concrete by either
V' or
E' alone, the functor
U:
Grph →
Set2 sending object
G to the pair of sets (
Grph(
V',
G),
Grph(
E',
G)) and morphism
h:
G →
H to the pair of functions (
Grph(
V',
h),
Grph(
E',
h)) is faithful. That is, a morphism of graphs can be understood as a
pair of functions, one mapping the vertices and the other the edges, with application still realized as composition but now with multiple sorts of
generalized elements. This shows that the traditional concept of a concrete category as one whose objects have an underlying set can be generalized to cater for a wider range of toposes by allowing an object to have multiple underlying sets, that is, to be multisorted.
See also
- Background and genesis of topos theory
This page gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given...
- Category theory
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
- Intuitionistic type theory
Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism. Intuitionistic type theory was introduced by Per Martin-Löf, a Swedish mathematician and philosopher,...