Vipul at 01:18, 10 December 2008

2008-12-10T01:18:27Z

← Older revision		Revision as of 01:18, 10 December 2008
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	==Definition==		==Definition==

	Suppose <math>\mathcal{C}</math> is a [[category]] and <math>A \in \operatorname{Ob}\mathcal{C}</math>. The '''covariant Hom functor''' corresponding to <math>A</math> is defined as a covariant functor from <math>\mathcal{C}</math> to the [[category of sets]] given as follows:		Suppose <math>\mathcal{C}</math> is a [[locally small category]] and <math>A \in \operatorname{Ob}\mathcal{C}</math>. The '''covariant Hom functor''' corresponding to <math>A</math> is defined as a covariant functor from <math>\mathcal{C}</math> to the [[category of sets]] given as follows:

	* '''On objects''': An object <math>B \in \operatorname{Ob}\mathcal{C}</math> is mapped to <math>\mathcal{C}(A,B)</math>.		* '''On objects''': An object <math>B \in \operatorname{Ob}\mathcal{C}</math> is mapped to <math>\mathcal{C}(A,B)</math>.

Vipul: New page: ==Definition== Suppose $\mathcal{C}$ is a category and $A \in \operatorname{Ob}\mathcal{C}$ . The '''covariant Hom functor''' corresponding to $A$ is d...

2008-12-09T23:33:23Z

New page: ==Definition== Suppose <math>\mathcal{C}</math> is a category and <math>A \in \operatorname{Ob}\mathcal{C}</math>. The '''covariant Hom functor''' corresponding to <math>A</math> is d...

New page

==Definition==

Suppose <math>\mathcal{C}</math> is a [[category]] and <math>A \in \operatorname{Ob}\mathcal{C}</math>. The '''covariant Hom functor''' corresponding to <math>A</math> is defined as a covariant functor from <math>\mathcal{C}</math> to the [[category of sets]] given as follows:

* '''On objects''': An object <math>B \in \operatorname{Ob}\mathcal{C}</math> is mapped to <math>\mathcal{C}(A,B)</math>.
* '''On morphisms''': Given objects <math>B,C \in \operatorname{Ob}\mathcal{C}</math> and an element <math>f \in \mathcal{C}(A,B)</math>, <math>f</math> gets sent to the map <math>\mathcal{F}f: \mathcal{C}(A,B) \to \mathcal{C}(A,C)</math> defined by:

<math>\mathcal{F}f = g \mapsto f \circ g </math>.

A set-valued functor is termed [[representable functor|representable]] if it is naturally isomorphic to a covariant Hom functor.

==Facts==

* The covariant Hom functor is actually a functor. {{proofat|[[Covariant Hom functor is functor]]}}
* A morphism between two objects in <math>\mathcal{C}</math> induces a natural transformation between the corresponding covariant Hom functors. {{proofat|[[Morphism induces natural transformation of induced covariant Hom functors]]}}

Covariant Hom functor - Revision history

Vipul at 01:18, 10 December 2008

Vipul: New page: ==Definition== Suppose \mathcal{C} is a category and A \in \operatorname{Ob}\mathcal{C}. The '''covariant Hom functor''' corresponding to A is d...

Vipul: New page: ==Definition== Suppose $\mathcal{C}$ is a category and $A \in \operatorname{Ob}\mathcal{C}$ . The '''covariant Hom functor''' corresponding to $A$ is d...