Category theory

Category theory

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": Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ :\text{Mor}_{\mathscr{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathscr{C}} and Failed to parse (unknown function "\mathscr"): {\displaystyle \mathscr{D}} are two

categories, then there is a category consisting of all contravarient functors from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathscr{C}} to Failed to parse (unknown function "\mathscr"): {\displaystyle \mathscr{D}} , where morphisms are natural transformations.

1. The category of schemes is one of the principal objects of study