Category theory: Difference between revisions
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imported>Giovanni Antonio DiMatteo No edit summary |
imported>David E. Volk m (subpages, move categories to metadata) |
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Category theory | '''Category theory''' | ||
==Definition== | ==Definition== | ||
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#[[Category of functors|The category of functors]]: if <math>C</math> and <math>D</math> are two categories, then there is a category consisting of all contravarient functors from <math>C</math> to <math>D</math>, where morphisms are [[Category of functors|natural transformations]]. | #[[Category of functors|The category of functors]]: if <math>C</math> and <math>D</math> are two categories, then there is a category consisting of all contravarient functors from <math>C</math> to <math>D</math>, where morphisms are [[Category of functors|natural transformations]]. | ||
#[[Scheme|The category of schemes]] is one of the principal objects of study | #[[Scheme|The category of schemes]] is one of the principal objects of study | ||
Revision as of 01:26, 13 January 2008
Category theory
Definition
A category consists of the following data:
- A class of "objects," denoted
- For objects , a set such that is empty if and
together with a "law of composition": (which we denote by ) having the following properties:
- Associativity: whenever the compositions are defined
- Identity: for every object there is an element such that for all , and .
Examples
- The category of sets:
- The category of topological spaces:
- The category of functors: if and are two categories, then there is a category consisting of all contravarient functors from to , where morphisms are natural transformations.
- The category of schemes is one of the principal objects of study