Category theory: Difference between revisions

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imported>Giovanni Antonio DiMatteo
imported>Giovanni Antonio DiMatteo
Line 13: Line 13:
#[[Set|The category of sets]]:
#[[Set|The category of sets]]:
#[[Topological space|The category of topological spaces]]:
#[[Topological space|The category of topological spaces]]:
#[[Category of functors|The category of functors]]: if <math>\mathscr{C}</math> and <math>\mathscr{D}</math> are two
#[[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>\mathscr{C}</math>
categories, then there is a category consisting of all contravarient functors from <math>C</math>
to <math>\mathscr{D}</math>, where morphisms are [[Category of functors|natural transformations]].
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 08:59, 1 January 2008

Category theory

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.

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