Category theory

Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. It is half-jokingly known as "abstract nonsense".

See list of category theory topics for a breakdown of the relevant Wikipedia pages.

Table of contents

1 Background 2 Historical notes 3 Categories 3.1 Definition 3.2 Examples 4 Types of morphisms 5 Special objects 6 Functors 6.3 Definition 6.4 Examples 7 Natural transformations and natural isomorphisms 7.5 Definition 7.6 Examples 8 Equivalence of categories 8.7 Definition 8.8 Examples 9 Further concepts and results 10 Types of categories 11 Literature

Background

A category attempts to capture the essence of a class of related mathematical objects, for instance the class of groups. Instead of focusing on the individual objects (groups) as has been done traditionally, the morphisms, i.e. the structure preserving maps between these objects, are emphasized. In the example of groups, these are the group homomorphisms. Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second. Very commonly, certain "natural constructions", such as the fundamental group of a topological space, can be expressed as functors. Furthermore, different such constructions are often "naturally related" which leads to the concept of natural transformation, a way to "map" one functor to another. Throughout mathematics, one encounters "natural isomorphisms", things that are (essentially) the same in a "canonical way". This is made precise by special natural transformations, the natural isomorphisms.

Historical notes

Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders MacLane in 1945. Initially, the notions were applied in topology, especially algebraic topology, as an important part of the transition from homology (an intuitive and geometric concept) to homology theory, an axiomatic approach. It has been claimed, for example by or on behalf of Ulam, that comparable ideas were current in the later 1930s in the Polish school.

Eilenberg/MacLane have said that their goal was to understand natural transformations; in order to do that, functors had to be defined; and to define functors one needed categories.

The subsequent development of the theory was powered first by the computational needs of homological algebra; and then by the axiomatic needs of algebraic geometry, the field most resistant to the Russell-Whitehead view of united foundations. General category theory - an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic - came later; it is now applied throughout mathematics.

Special categories called topoi can even serve as an alternative to axiomatic set theory as the foundation of mathematics. These broadly-based foundational applications of category theory are contentious; but they have been worked out in quite some detail, as a commentary on or basis for constructive mathematics. One can say, in particular, that axiomatic set theory still hasn't been replaced by the category-theoretic commentary on it, in the everyday usage of mathematicians. The idea of bringing category theory into earlier, undergraduate teaching (signified by the difference between the Birkhoff- Mac Lane and later Mac Lane-Birkhoff abstract algebra texts) has hit noticeable opposition.

Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus. At the very least, the use of category theory language allows one to clarify what exactly these related areas have in common.

Categories

Definition

A category consists of:

• a class of things called objects.
• for every two objects A and B a set Mor(A,B) of things called morphisms from A to B. If f is in Mor(A,B), we write f : A -> B.
• for every three objects A, B and C a binary operation Mor(A,B) x Mor(B,C) -> Mor(A,C) called composition of morphisms. The composition of f : A -> B and g : B -> C is written as g o f or gf. (Some authors write it as fg.)

such that the following axioms hold:

• (associativity) if f : A -> B, g : B -> C and h : C -> D then h o (g o f) = (h o g) o f, and
• (identity) for every object X there exists a morphism id_X : X -> X called the identity morphism for X, such that for every morphism f : A -> B we have id_B o f = f = f o id_A

From these axioms, one can prove that there is exactly one identity morphism for every object.

If the class of objects is actually a set, the category is said to be small. Many important categories are not small.

Examples

Each category is presented in terms of its objects and its morphisms.

• The category Grp consisting of all groups together with their group homomorphisms.
• The category Vect_K of all vector spaces over the field K together with their K-linear maps.
• The category Set of all sets together with functions between sets.
• The category Top of all topological spaces with continuous functions.
• Any partially ordered set (P, ≤) forms a small category, where the objects are the members of P, and the morphisms are arrows pointing from x to y precisely when x ≤ y.
• Any monoid forms a small category with a single object x, and where every element of the monoid is a morphism from x to x (the monoid operation yields the categorical composition of morphisms). In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may be generalized for categories.
• Any directed graph can be considered as a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph. Composition of morphisms is concatenation of paths.
• If I is a set, the discrete category on I is the small category which has the elements of I as objects and only the identity morphisms as morphisms.
• Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted by C^op.
• If C and D are categories, one can form the product category C x D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.

Types of morphisms

A morphism f : A -> B is called a

• monomorphism if fg₁ = fg₂ implies g₁ = g₂ for all morphisms g₁, g₂ : X -> A.
• epimorphism if g₁f = g₂f implies g₁ = g₂ for all morphisms g₁, g₂ : B -> X.
• isomorphism if there exists a morphism g : B -> A with fg = id_B and gf = id_A.
• automorphism if f is an isomorphism and A = ''B'\'.
• endomorphism if A = B.

Relations among morphisms (such as fg = h) can most conveniently be represented with commutative diagrams, where the objects are represented as points and the morphisms as arrows.

Special objects

An object A of the category C is called

• initial, if for every object B there is a unique morphism A -> B. For example, the empty set is an initial object in the category of sets.
• terminal, if for every object B there is a unique morphism B -> A. For example, every singleton (set with one element) is a terminal object in the category of sets.

Functors

Functors are structure-preserving maps between categories.

Definition

A (covariant) functor F from the category C to the category D

• associates to each object X in C an object F(X) in D;
• associates to each morphism f:X->Y a morphism F(f):F(X)->F(Y)

such that the following two properties hold:

• F(id_X) = id_F(X) for every object X in C.
• F(g o f) = F(g) o F(f) for all morphisms f : X -> Y and g : Y -> Z.

A contravariant functor F from C to D is a functor that "turns morphisms around" (i.e. if f:X->Y is a morphism in C, then F(f):F(Y)->F(X)); the quickest way to define a contravariant functor is as a covariant functor between C^op and D.

An important consequence of the functor axioms is this: if f is an isomorphism in C, then F(f) is an isomorphism in D.

Examples

Dual vectorspace: an example of a contravariant functor from the category of all real vector spaces to the category of all real vector spaces is given by assigning to every vector space its dual space and to every linear map its dual or transpose.

Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X -> Y induces an algebra homomorphism C(f) : C(Y) -> C(X) by the rule C(f)(φ) = φ o f for every φ in C(Y).

Homomorphism groups: to every pair A, B of abelian groups and can assign the abelian group Hom(A,B) consisting of all group homomorphisms from A to B. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor Ab^op x Ab -> Ab (where Ab denotes the category of abelian groups with group homomorphisms). If f : A₁ -> A₂ and g : B₁ -> B₂ are morphisms in Ab, then the group homomorphism Hom(f,g) : Hom(A₂,B₁) -> Hom(A₁,B₂) is given by φ |-> g o φ o f.

\Forgetful functors: the functor F : Ring -> Ab which maps a ring to its underlying abelian additive group. Morphisms in Ring (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).

Tensor products: If C denotes the category of vectorspaces over a fixed field, with linear maps as morphisms, then the tensor product VW defines a functor C x C -> C which is covariant in both arguments.

Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor.

Fundamental group: Consider the category of topological spaces with distinguished points. The objects are pairs (X,x), where X is a topological space and x is an element of X. A morphism from (X,x) to (Y,y) is given by a continuous map f : X -> Y with f(x) = y.

For every topological space with distinguished point (X,x), we will define a fundamental group. This is going to be a functor from the category of topological spaces with distinguished points to the category of groups.

Let f be a continuous function from the unit interval [0,1] into X so that f(0) = f(1) = x. (Equivalently, f is a continuous map from the unit circle in the complex plane so that f(1) = x.) We call such a function a loop in X. If f and g are loops in X, we can glue them together by defining h(t) = f(2t) when t is in [0,0.5] and h(t) = g(2(t - 0.5)) when t is in [0.5,1]. It is easy to check that h is again a loop. If there is a continuous map F(x,t) from [0,1] × [0,1] to X so that f(t) = F(0,t) is a loop and g(t) = F(1,t) is also a loop then f and g are said to be equivalent. It can be checked that this defines an equivalence relation. Our composition rule survives this process. Now, in addition, we can see that we have an identity element e(t) = x (a constant map) and further that every loop has an inverse. Indeed, if f(t) is a loop then f(1 - t) is its inverse. The set of equivalence classes of loops thus forms a group (the fundamental group of X). One may check that the map from the category of topological spaces with a distinguished point to the category of groups is functorial: a topological (homo/iso)morphism will naturally correspond to a group (homo/iso)morphism.

Representable functors: If C is a category and U an object in C, then F(X) = Mor_C(U,X) defines a covariant functor form C to Set. Functors like these are called representable, and a major goal in many settings is to determine whether a given functor is representable.

Universal constructions: Functors are often defined by universal properties; examples are the tensor product discussed above, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of adjoint functors.

Pre-Sheaves: If X is a topological space, then the open sets in X can be considered as the objects of a category CX; there is a morphism from U to V if and only if U is a subset of V. In itself, this category is not very exciting, but the functors from CX^op into other categories, the so-called pre-sheaves on X, are interesting. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a pre-sheaf of algebras on X.

This motivating example is generalized by considering pre-sheaves on arbitrary categories: a pre-sheaf on C is a functor defined on C^op. The Yoneda lemma explains that often a category C can be extended by considering a category of pre-sheaves on C.

Category of small categories: The category Cat has the small categories as objects, and the functors between them as morphisms.

Natural transformations and natural isomorphisms

A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions. Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.

Definition

If F and G are (covariant) functors between the categories C and D, then a natural transformation from F to G associates to every object X in C a morphism η_X : F(X) -> G(X) in D such that for every morphism f : X -> Y in C we have η_Y o F(f) = G(f) o η_X; this means that the following diagram is commutative:

The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that η_X is an isomorphism for every object X in C.

Examples

If K is a field, then for every vector space V over K we have a "natural" injective linear map V -> V^** from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps form a natural transformation from the identity functor to the double dual functor. If we restrict to finite-dimensional vector spaces, we even obtain a natural isomorphism. "Every finite-dimensional vector space is naturally isomorphic to its double dual."

Consider the category Ab of abelian groups and group homomorphisms. For all abelian groups X, Y and Z we have a group isomorphism

Hom(X, Hom(Y, Z)) -> Hom(XY, Z).

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab^op x Ab^op x Ab -> Ab.

Equivalence of categories

Intuitively, two equivalent categories cannot be distinguished from the standpoint of category theory.

Definition

Two categories C and D are called equivalent if there exist functors F : C -> D and G : D -> C such that FG is naturally isomorphic to I_D (where I_D denotes the identity functor D -> D which assigns every object to itself and every morphism to itself) and GF is naturally isomorphic to I_C.

Examples

One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings. This is in fact a contravariant equivalence or duality, meaning that the two functor F and G are contravariant functors. The functor G associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring. The functor F associates to every affine scheme its ring of global sections.

Another important duality occurs in functional analysis: the category of commutative C*-algebras with identity is contravariantly equivalent to the category of compact Hausdorff spaces. Under this duality, every compact Hausdorff space X is associated with the algebra of continuous complex-valued functions on X, and every commutative C*-algebra is associated with the space of its maximal ideals. This is the Gelfand representation.

Further concepts and results

The definitions of categories and functors provide only the very basics of categorical algebra. Additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.

• The functor category D^C has as objects the functors from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories.
• Universal properties can be used to define a great number of constructions (and hence: functors) throughout mathematics.
• Limits and colimits are defined by special univeral properties, and generalize products (of sets, of topologies, of partial orders, ...).
• Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; it can be seen as a more abstract and powerful view on universal properties.

Types of categories

• In many categories, the morphism sets Mor(A,B) are not just sets but actually abelian groups, and the composition of morphisms is compatible with these group structures, i.e. is bilinear. Such a category is called preadditive. If furthermore the category has all finite products and coproducts, it is called an additive category. If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category. A typical example of an abelian category is the category of abelian groups.
• A category is called complete if all limits in it exist. The categories of sets, abelian groups and topological spaces are complete.
• A category is called cartesian closed if it has finite direct products and a morphism defined on a finite product can always be represented by a morphism defined on just one of the factors.
• A topos is a certain type of cartesian closed category in which all of mathematics can be formulated (just like classically all of mathematics is formulated in the category of sets). A topos can also be used to represent a logical theory.
• A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group actions and equivalence relations.

Literature

• William Lawvere and Steve Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997.
• Saunders Mac Lane: Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer 1998
• Francis Borceux: Handbook of Categorical Algebra, volumes 50-52 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1994.