Category theory: Difference between revisions
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imported>David E. Volk m (subpages, move categories to metadata) |
imported>Peter Lyall Easthope (Drafted the introductory paragraph suggested by Noel Chiappa.) |
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'''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== | ==Definition== |
Revision as of 14:52, 18 May 2008
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:
- 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