Category theory

Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. Although originally mainly developed in the context of algebraic geometry, [algebraic topology]? and [universal algebra]?, it is now used in various branches of mathematics and is by some even considered as a better foundation of mathematics than set theory.

A category attempts to capture the essense of a class of related mathematical structures, 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 would be 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, 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.

Categories

Definition:

A category consists of:

• a collection of things called objects,
• a collection of things called morphisms or arrows,
• a relation over morphisms and pairs of objects called typing of the morphisms
  (if a morphism f is associated with the pair (A,B) then this is written as f : A -> B),
• a binary partial operation on the collection of objects called composition
  (the composition of f and g is written as f o g, but also often as g ; f or simply as f g), and
• for every object A a distinguished morphism id_A called the identity on A,

and the following typing axioms hold:

• (unique type) if f : A -> B and f : C -> D then A = C and B = D
• (composition type) if f : A -> B and g : B -> C then g o f : A -> C
• (identity type) id_A : A -> A

and the following equality axioms hold:

• (associativity) if f : C -> D, g : B -> C and h : A -> B then f o (g o h) = (f o g) o h, and
• (identity) if f : A -> B then id_B o f = f = f o id_A.

If the collections in question are sets, 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.

• Mathematical groups together with their group homomorphisms.
• Vector spaces together with their linear maps.
• Sets together with functions on those sets.
• Hausdorff spaces with continuous functions.
• Small categories with functors as the morphisms.
• Any partially ordered set (P, <=) forms a category, where the objects are the members of P are the objects, and the morphisms are arrows pointing from x to y precisely when x <= y in P.
• Any monoid forms a category with a single object x, and where every element of the monoid is a morphism from x to x (the categorical composition of morphisms gives the composition of elements in the monoid). In fact, one may view categories as generalizations of monoids; several definitions and theorems about monoids may also be given for categories.
• Any [directed graph]? can be considered as a 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.
• Any category can itself be considered as a 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.

Functors

Functors are structure-preserving maps between categories.

Definition

A functor is a map F between categories C and D such that

• each object X in C maps to an object F(X) in D;
• each morphism f:X->Y maps to a morphism F(f):F(X)->F(Y) such that identity functions map to identifiy functions, and the law of composition is preserved.

Examples

• Forgetful functors, e.g. the functor F : Ring -> AbGroup which maps a ring to its underlying abelian? additive group. Morphisms in Ring (ring homomorphisms) become morphisms in AbGroup (abelian group homomorphisms).

Here we give a nontrivial example of a functor. There is the category of Hausdorff topological spaces. A topological space is a set together with a family of open sets. A morphism of topological spaces is a continuous function, that is, a function f from X to Y (topological spaces) for which the preimage? of any open set is also open. An isomorphism of topological spaces is a continuous, surjective function with an inverse that is also continuous. To say that the topological space X is Hausdorff is to say that, if x, y are in X, then there are open sets U, V with x in U and y in V and U and V do not intersect.

There is also the category of groups. A group G is a set together with a multiplication law, a unit and inverses. That is, if x, y are in G, then the product x · y is also in G. Further, there is a distinguished element e in G so that, for any x in G, ex = xe = x. Lastly, for each element x in G, there is an element y in G so that xy = yx = e. Morphisms or homomorphisms are functions f such that f(xy)=f(x)f(y). Isomorphisms are surjective, injective homomorphisms.

Given a topological space X and a distinguished point x in X, we can create a group. 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) are also loops then f and g are said to be equivalent. It can be checked that this defines an honest to goodness 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 Hausdorff 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.

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Related topics

• monic and epic morphisms (or mono and epi)
• commutative diagrams
• functors
• groupoids
• the Yoneda lemma
• [Natural transformation]?s

• [History of Category Theory]?
• [Reception of Category Theory]?