Functor: Difference between revisions
(New page: ==Definition== Suppose <math>\mathcal{C},\mathcal{D}</math> are categories. A '''functor''' <math>\mathcal{F}</math> from <math>\mathcal{C}</math> to <math>\mathcal{D}</math>...) |
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==Definition== | ==Definition== | ||
Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]]. A '''functor''' <math>\mathcal{F}</math> from <math>\mathcal{C}</math> to <math>\mathcal{D}</math> comprises the following data: | Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]]. A '''functor''' (also called '''covariant functor''') <math>\mathcal{F}</math> from <math>\mathcal{C}</math> to <math>\mathcal{D}</math> comprises the following data: | ||
* A mapping <math>\mathcal{F}: \operatorname{Ob}\mathcal{C} \to \operatorname{Ob}\mathcal{D}</math>. | * A mapping <math>\mathcal{F}: \operatorname{Ob}\mathcal{C} \to \operatorname{Ob}\mathcal{D}</math>. | ||
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* For any <math>A \in \mathcal{C}</math>, <math>\mathcal{F}(\operatorname{id}_A) = \operatorname{id}_{F(A)}</math>. | * For any <math>A \in \mathcal{C}</math>, <math>\mathcal{F}(\operatorname{id}_A) = \operatorname{id}_{F(A)}</math>. | ||
* For any <math>A,B,C \in \mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, we have <math>F(g \circ f) = Fg \circ Ff</math>. | * For any <math>A,B,C \in \mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, we have <math>F(g \circ f) = Fg \circ Ff</math>. | ||
There is a related notion of [[contravariant functor]]. Note that a contravariant functor between two categories is ''not'' a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term ''covariant functor''. | |||
Revision as of 23:18, 9 December 2008
Definition
Suppose are categories. A functor (also called covariant functor) from to comprises the following data:
- A mapping .
- For any , a mapping .
satisfying the following condition:
- For any , .
- For any , and , we have .
There is a related notion of contravariant functor. Note that a contravariant functor between two categories is not a functor between the categories. To emphasize that a functor is a functor and not a contravariant functor, we use the term covariant functor.