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 .
of 
.
of 
of Which satisfies the conditions:
for all 
.
for all morphisms and 
of .
for all 
.
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.
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