Vipul at 23:21, 9 December 2008

2008-12-09T23:21:43Z

← Older revision		Revision as of 23:21, 9 December 2008
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	* 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>.
	* For every <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a mapping <math>\mathcal{F}: \mathcal{C}(A,B) \to \mathcal{D}(\mathcal{F}B,\mathcal{F}A)</math>.		* For every <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a mapping <math>\mathcal{F}: \mathcal{C}(A,B) \to \mathcal{D}(\mathcal{F}B,\mathcal{F}A)</math>.

			satisfying the following conditions:

			* It preserves the identity map: For any <math>A \in \operatorname{Ob}\mathcal{C}</math>, <math>\mathcal{F}(\operatorname{id}_A) = \operatorname{id}_{\mathcal{F}A}</math>.
			* It preserves composition, albeit reversing the order of composition: For any <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, and <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C)</math>, we have <math>\mathcal{F}(g \circ f) = \mathcal{F}f \circ \mathcal{F}g</math>.

	===Definition in terms of the opposite category===		===Definition in terms of the opposite category===

Vipul: New page: ==Definition== ===Definition in basic terms=== Suppose $\mathcal{C}, \mathcal{D}$ are categories. A '''contravariant functor''' $\mathcal{F}:\mathcal{C} \to ...$

2008-12-09T23:17:14Z

New page: ==Definition== ===Definition in basic terms=== Suppose <math>\mathcal{C}, \mathcal{D}</math> are categories. A '''contravariant functor''' <math>\mathcal{F}:\mathcal{C} \to ...

New page

==Definition==

===Definition in basic terms===

Suppose <math>\mathcal{C}, \mathcal{D}</math> are [[category|categories]]. A '''contravariant functor''' <math>\mathcal{F}:\mathcal{C} \to \mathcal{D}</math> is defined by the following data:

* A mapping <math>\mathcal{F}: \operatorname{Ob}\mathcal{C} \to \operatorname{Ob}\mathcal{D}</math>.
* For every <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a mapping <math>\mathcal{F}: \mathcal{C}(A,B) \to \mathcal{D}(\mathcal{F}B,\mathcal{F}A)</math>.

===Definition in terms of the opposite category===

Suppose <math>\mathcal{C},\mathcal{D}</math> are [[category|categories]]. A '''contravariant functor''' from <math>\mathcal{C}</math> to <math>\mathcal{D}</math> can be defined as:

* A [[defining ingredient::functor]] from <math>\mathcal{C}^{op}</math> (the [[defining ingredient::opposite category]] to <math>\mathcal{C}</math>) to <math>\mathcal{D}</math>.
* A [[functor]] from <math>\mathcal{C}</math> to <math>\mathcal{D}^{op}</math> (the [[opposite category]] to <math>\mathcal{D}</math>).

A functor in the usual sense of the word is sometimes termed a '''covariant functor''' to contrast it with a contravariant functor.

Contravariant functor - Revision history

Vipul at 23:21, 9 December 2008

Vipul: New page: ==Definition== ===Definition in basic terms=== Suppose \mathcal{C}, \mathcal{D} are categories. A '''contravariant functor''' \mathcal{F}:\mathcal{C} \to ...

Vipul: New page: ==Definition== ===Definition in basic terms=== Suppose $\mathcal{C}, \mathcal{D}$ are categories. A '''contravariant functor''' $\mathcal{F}:\mathcal{C} \to ...$