Category theory

Category theory

Definition

A category consists of the following data:

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

together with a "law of composition": Failed to parse (unknown function "\mathscr"): {\displaystyle \circ :\text{Mor}_{\mathscr{C}}(B,C)\times\text{Mor}_{\mathscr{C}}(A,B)\to \text{Mor}_{\mathscr{C}}(A,C)} (which we denote by $(g,f)\mapsto g\circ f$) which is

1. 1. Associative: $(h\circ g)\circ f=h\circ (g\circ f)$ whenever the compositions are defined
   2. Having identity: for every object Failed to parse (unknown function "\mathscr"): {\displaystyle A\in ob(\mathscr{C})} there is an element $id_{A}$ such that for all Failed to parse (unknown function "\mathscr"): {\displaystyle f\in\text{Mor}_{\mathscr{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 (unknown function "\mathscr"): {\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 (unknown function "\mathscr"): {\displaystyle \mathscr{C}} to 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{D}} , where morphisms are natural transformations.

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