# Difference between revisions of "Functor"

(New page: A '''functor''' is a type of map between two categories. More precisely, a functor <math>F:\mathcal{C} \to \mathcal{D}</math> is a mapping which * sends eve...) |
(talked about contravariant functors) |
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* <math>F(1_X) = 1_{F(X)}</math> for all <math>X\in \text{Ob}(\mathcal{C})</math>. | * <math>F(1_X) = 1_{F(X)}</math> for all <math>X\in \text{Ob}(\mathcal{C})</math>. | ||

* <math>F(g\circ f) = F(g)\circ F(f)</math> for all morphisms <math>f:X\to Y</math> and <math>g:Y \to Z</math> of <math>\mathcal{C}</math>. | * <math>F(g\circ f) = F(g)\circ F(f)</math> for all morphisms <math>f:X\to Y</math> and <math>g:Y \to Z</math> of <math>\mathcal{C}</math>. | ||

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+ | A '''contravariant functor''' a mapping satisfying the same properties as above, except that <math>F(f)</math> is a morphism from <math>F(Y)</math> to <math>F(X)</math>, and instead of having <math>F(g\circ f) = F(g)\circ F(f)</math> we have <math>F(g\circ f) = F(f)\circ F(g)</math>. Alternatively, we can define a contravariant functor as an ordinary functor <math>F:\mathcal{C}^{op}\to \mathcal{D}</math>, where <math>\mathcal{C}^{op}</math> is the [[opposite category]] of <math>\mathcal{C}</math>. We sometimes call our original type of functors '''covariant functor''' to distinguish them from contravariant functors. | ||

{{stub}} | {{stub}} | ||

[[Category:Category theory]] | [[Category:Category theory]] |

## Latest revision as of 21:42, 2 September 2008

A **functor** is a type of map between two categories.

More precisely, a functor is a mapping which

- sends every object of to and object of .
- sends every morphism of to a morphism of .

Which satisfies the conditions:

- for all .
- for all morphisms and of .

A **contravariant functor** a mapping satisfying the same properties as above, except that is a morphism from to , and instead of having we have . Alternatively, we can define a contravariant functor as an ordinary functor , where is the opposite category of . We sometimes call our original type of functors **covariant functor** to distinguish them from contravariant functors.

*This article is a stub. Help us out by expanding it.*