# Categories and Functors
A category consists of the following data:
- A set of wholse elements are called objects of ;
- A collection of sets , one for each ordered pair of objects , whose elements are called morphisms (from to ); they are denoted ;
- A collection of mappings
one for each ordered triple of objects . Any mapping in this collection associates to a pair , , a morphism from to , denoted , and called the composition or product of and
This data should satisfy the following conditions:
- Any morphism uniquely determines such that . In other words, the sets are pairwise disjoint.
- For any there exists the identity morphism of ; it is determined uniquely by the conditions , wherever these compositions are defined.
- The composition of morphisms is associateive
for any , , .
Sometimes it is abbreviated rather thatn , and or instead of . A morphism may sometimes be called an arrow starting at and ending at . The set is denoted .
# Category of Sets and Categories of Sets with a Structure
An important class of categories is formed by categories whose objects are sets with some additional structure, and whose morphisms are maps respecting this structure.
- , the category of topological spaces with continuous maps between them;
- , the category of smooth manifolds and smooth maps;
- , the category of abelian groups and homomorphisms;
- -mod, the category of left-modules over some fixed ring ;
- , the category of groups and homomorphisms.
# More Examples
In this group of examples, the objects of the categories are the same as in the list above, i.e. sets with additional structure, but here we define different morphims.
- The category :
- The category of relations :
The composition of and is defined as follows:
The identity morphism is the diagonal
- The category of addititve relations:
The composition of morphisms and the identity morphism are defined the same way as in the previous example.
- The category of a partially ordered set :
The composition of morphisms and the identity morphism are defined in the only possible way. An important special case of is:
- The category . Let be a topological space. Define
A covariant functor from a category to a category (notation: ) consists of the following data:
- A mapping ;
- A mapping such that for we have that .
This data should satisfy the following conditions: for any for which is defined. In particular, .
Consider the mapping ,
Given any category , we can construct the opposite category of , denoted . It has the same objects as , but for any two , .
A contravariant functor from a category to a category is a covariant functor from to .
- Any presheaf of abelian groups on a topological space is a functor
- Let be a partially ordered set, be the corresponding category. A functor is a family of abelian groups and of maps , one for each ordered pair . These maps should satisfy the conditions for , . Such families usually appear as raw material for inductive and/or projective limits.
- By simply forgetting one or more structures on an object, we obtain many functors for the initial category. For example,
are functors that simply forget the topological, differentiable, abelian group, and group structures of the objects in the initial category, respectively. Since these functors "forget" some or all of the structure(s) of the object in the initial category, these are examples of forgetful functors.
Let be two functors from to . A morphism of functors from to (notation: ) is a family of morphisms in :
one for each , satisfying the following condition: for any morphism in , the diagram (insert diagram) is commutative.
The composition of morphisms of functors, as well as the identity morphism of functors, are defined in the obvious way. Thus, functors from to form a category in their own right, which is usually denoted .
A category is said to be a subcategory of a category if
- for any ;
- The composition of morphisms in coincide with their composition in , for the identity morphism in coincides with the identity morphism in .
A subcategory is said to be full if = for any .
A functor is said to be faithful if for any the map
is injective, and full if this map is surjective.
An object of a category is said to be an initial object if is a one-element set of any . Similarly, an object is said to be a final object in if is a one-element set for any .
Both initial and final objects are determined uniquely up to isomorphism, if they exist.
# To Do
- Isomorphisms of categories
- Equivalence of categories
- Quasi-inverses of functors