Category theory: Difference between revisions

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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

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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:

  1. A class of "objects," denoted
  2. For objects , a set such that is empty if and

together with a "law of composition": (which we denote by ) having the following properties:

    1. Associativity: whenever the compositions are defined
    2. Identity: for every object there is an element such that for all , and .

Examples

  1. The category of sets:
  2. The category of topological spaces:
  3. 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.
  4. The category of schemes is one of the principal objects of study