Category theory

Category theory

A natural language has nouns and verbs. High school mathematics introduces sets and functions acting on them. A computer program can act upon an entity of information, producing another entity. A vector can be subjected to a linear transformation. From observations such as these, two mathematical concepts are distilled: the object and the map or morphism.

Definition

A category consists of the following data:

1. A class of "objects," denoted $ob(C)$
2. For objects $A,B,C\in ob(C)$, a set ${\text{Mor}}_{C}(A,B)$ such that ${\text{Mor}}_{C}(A,B)\cap {\text{Mor}}_{C}(A',B')$ is empty if $A\neq A'$ and $B\neq B'$

together with a "law of composition": $\circ :{\text{Mor}}_{C}(B,C)\times {\text{Mor}}_{C}(A,B)\to {\text{Mor}}_{C}(A,C)$ (which we denote by $(g,f)\mapsto g\circ f$) having the following properties:

1. 1. Associativity: $(h\circ g)\circ f=h\circ (g\circ f)$ whenever the compositions are defined
   2. Identity: for every object $A\in ob(C)$ there is an element $id_{A}$ such that for all $f\in {\text{Mor}}_{C}(A,B)$, $id_{B}\circ f=f$ and $f\circ id_{A}=f$.

Examples

1. The category of sets:
2. The category of topological spaces:
3. The category of functors: if $C$ and $D$ are two categories, then there is a category consisting of all contravarient functors from $C$ to $D$, where morphisms are natural transformations.
4. The category of schemes is one of the principal objects of study