Category theory
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Category theory
Definition
A category consists of the following data:
- A class of "objects," denoted 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 ob(\mathscr{C})}
- 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 and
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 ) which is
- Associative: whenever the compositions are defined
- Having identity: for every object Failed to parse (unknown function "\mathscr"): {\displaystyle A\in ob(\mathscr{C})} there is an element such that for all Failed to parse (unknown function "\mathscr"): {\displaystyle f\in\text{Mor}_{\mathscr{C}}(A,B)} , and .
Examples
- The category of sets:
- The category of topological spaces:
- 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 (unknown function "\mathscr"): {\displaystyle \mathscr{D}} , where morphisms are natural transformations.
- The category of schemes is one of the principal objects of study