In category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, an abstract branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, an initial object of a category

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...

 C is an object I in C such that for every object X in C, there exists precisely one morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

 I → X. The dual

Dual (category theory)

In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

 notion is that of a terminal object (also called terminal element): T is terminal if for every object X in C there exists a single morphism X → T. Initial objects are also called coterminal or universal, and terminal objects are also called final.

If an object is both initial and terminal, it is called a zero object or null object.

Examples

• The empty set
  Empty set
  In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
  
   is the unique initial object in 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...
  
  ; every one-element set (singleton) is a terminal object in this category; there are no zero objects.
• Similarly, the empty space is the unique initial object in the category of topological spaces
  Category of topological spaces
  In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
  
  ; every one-point space is a terminal object in this category.
• In the category of non-empty sets, there are no initial objects. The singletons are not initial: while every non-empty set admits a function
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
   from a singleton, this function is in general not unique.
• In the category of groups
  Category of groups
  In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
  
  , any trivial group
  Trivial group
  In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
  
   is a zero object. The same is true for the categories of 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, modules
  Module (mathematics)
  In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
  
   over a ring, and vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
  s over a field. This is the origin of the term "zero object".
• In the category of semigroup
  Semigroup
  In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
  
  s, the empty semigroup
  Empty semigroup
  In mathematics, a semigroup with no elements is a semigroup in which the underlying set is the empty set. Many authors do not admit the existence of such a semigroup. For them a semigroup is by definition a non-empty set together with an associative binary operation. However not all authors insist...
  
   is the unique initial object and any singleton semigroup is a terminal object. There are no zero objects. In the subcategory
  Subcategory
  In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...
  
   of monoid
  Monoid
  In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
  
  s, however, every trivial monoid (consisting of only the identity element) is a zero object.
• In the category Rel
  Category of relations
  In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.A morphism R : A → B in this category is a relation between the sets A and B, so ....
  
   of sets and relations, the empty set is the unique zero object.
• In the category of pointed set
  Pointed set
  In mathematics, a pointed set is a set X with a distinguished element x_0\in X, which is called the basepoint. Maps of pointed sets are those functions that map one basepoint to another, i.e. a map f : X \to Y such that f = y_0. This is usually denotedf : \to .Pointed sets may be regarded as a...
  
  s (whose objects are non-empty sets together with a distinguished element; a morphism from (A,a) to (B,b) being a function f : A → B with f(a) = b), every singleton is a zero object. Similarly, in the category of pointed topological spaces
  Pointed space
  In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
  
  , every singleton is a zero object.
• In the category of rings
  Category of rings
  In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...
  
   with unity and unity-preserving morphisms, the ring of integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s Z is an initial object. The trivial ring consisting only of a single element 0=1 is a terminal object. In the category of general rings with homomorphisms, the trivial ring is a zero object.
• In the category of fields, there are no initial or terminal objects. However, in the subcategory of fields of characteristic
  Characteristic (algebra)
  In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
  
   p, the prime field of characteristic p forms an initial object.
• Any partially ordered set
  Partially ordered set
  In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
  
   (P, ≤) can be interpreted as a category: the objects are the elements of P, and there is a single morphism from x to y if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   x ≤ y. This category has an initial object if and only if P has a least element; it has a terminal object if and only if P has a greatest element
  Greatest element
  In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
  
  .
• If a monoid
  Monoid
  In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
  
   is considered as a category with a single object, this object is neither initial or terminal unless the monoid is trivial, in which case it is both.
• In the category of graph
  Graph (mathematics)
  In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
  
  s, the null graph
  Null graph
  In the mathematical field of graph theory, the null graph may refer either to the order zero graph, or alternatively, to any edgeless graph .-Order zero graph:...
  
   (without vertices
  Vertex (graph theory)
  In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
  
   and edges) is an initial object. The graph with a single vertex and a single loop
  Loop (graph theory)
  In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no loops....
  
   is terminal. The category of simple graphs does not have a terminal object.
• Similarly, the category of all small categories with functor
  Functor
  In 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 when in the category of small categories....
  
  s as morphisms has the empty category as initial object and the category 1 (with a single object and morphism) as terminal object.
• Any topological space
  Topological space
  Topological 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...
  
   X can be viewed as a category by taking the open set
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
  s as objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. This is a special case of the case "partially ordered set", mentioned above. Take P:= the set of open subsets
• If X is a topological space (viewed as a category as above) and C is some small category, we can form the category of all contravariant functors from X to C, using natural transformation
  Natural transformation
  In 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. This category is called the category of presheaves on X with values in C. If C has an initial object c, then the constant functor which sends every open set to c is an initial object in the category of presheaves. Similarly, if C has a terminal object, then the corresponding constant functor serves as a terminal presheaf.
• In the category of scheme
  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...
  
  s, Spec(Z) the prime spectrum
  Spectrum of a ring
  In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
  
   of the ring of integers is a terminal object. The empty scheme (equal to the prime spectrum of the trivial ring) is an initial object.
• If we fix a homomorphism
  Group homomorphism
  In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
  
   f : A → B of 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, we can consider the category C consisting of all pairs (X, φ) where X is an abelian group and φ : X → A is a group homomorphism with f φ = 0. A morphism from the pair (X, φ) to the pair (Y, ψ) is defined to be a group homomorphism r : X → Y with the property ψ r = φ. The kernel
  Kernel (algebra)
  In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
  
   of f is a terminal object in this category; this is nothing but a reformulation of the universal property
  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...
  
   of kernels. With an analogous construction, the cokernel
  Cokernel
  In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
  
   of f can be seen as an initial object of a suitable category.
• In the category of interpretations of an algebraic
  Universal algebra
  Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
  
   model
  Model theory
  In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
  
  , the initial object is the initial algebra
  Initial algebra
  In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. The initiality provides a general framework for induction and recursion....
  
  , the interpretation that provides as many distinct objects as the model allows and no more.

Existence and uniqueness

Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if I₁ and I₂ are two different initial objects, then there is a unique isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The same is true for terminal objects.
For complete categories

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...

 there is an existence theorem for initial objects. Specifically, a (locally small) complete category C has an initial object if and only if there exist a set I (not a proper class) and an I-indexed family

Indexed family

In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers....

 (K_i) of objects of C such that for any object X of C there at least one morphism K_i → X for some i ∈ I.

Equivalent formulations

Terminal objects in a category C may also be defined as 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....

s of the unique empty diagram

Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...

 ∅ → C. Since the empty category is vacuously a discrete category

Discrete category

In mathematics, especially category theory, a discrete category is a category whose only morphisms are the identity morphisms. It is the simplest kind of category...

, a terminal object can be thought of as an empty

Empty product

In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...

 product

Product (category theory)

In category theory, the product of two objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces...

. Dually, an initial object is a colimit of the empty diagram ∅ → C and can be thought of as an empty

Empty sum

In mathematics, an empty sum, or nullary sum, is a summation involving no terms at all. The value of any empty sum of numbers is conventionally taken to be zero...

 coproduct

Coproduct

In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...

 or categorical sum.

It follows that any functor

Functor

In 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 when in the category of small categories....

 which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects. For example, the initial object in any concrete category

Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...

 with free object

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....

s will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor

Forgetful functor

In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output...

 to Set, preserves colimits).

Initial and terminal objects may also be characterized in terms of 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...

 and 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...

. Let 1 be the discrete category with a single object (denoted by •), and let U : C → 1 be the unique (constant) functor to 1. Then

• An initial object I in C is a universal morphism from • to U. The functor which sends • to I is left adjoint to U.
• A terminal object T in C is a universal morphism from U to •. The functor which sends • to T is right adjoint to U.

Relation to other categorical constructions

Many natural constructions in category theory can be formulated in terms of finding an initial or terminal object in a suitable category.

• A universal morphism from an object X to a functor U can be defined as an initial object in the comma category
  Comma category
  In mathematics, a comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W...
  
   (X ↓ U). Dually, a universal morphism from U to X is a terminal object in (U ↓ X).
• The 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....
  
   of a diagram
  Diagram (category theory)
  In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...
  
   F is a terminal object in Cone(F) the category of cones to F. Dually, a colimit of F is an initial object in the category of cones from F.
• A representation of a 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...
  
   F to Set is an initial object in the category of elements of F.

Other properties

• The endomorphism monoid of an initial or terminal object I is trivial: End(I) = Hom(I,I) = { id_I }.
• If a category C has a zero object 0 then for any pair of objects X and Y in C the unique composition X → 0 → Y is a zero morphism
  Zero morphism
  In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...
  
  from X to Y.