Covariant Hom functor: Difference between revisions

Latest revision as of 01:18, 10 December 2008

Definition

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

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

${\mathcal {F}}f=g\mapsto f\circ g$ .

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

Facts

The covariant Hom functor is actually a functor. For full proof, refer: Covariant Hom functor is functor
A morphism between two objects in ${\mathcal {C}}$ induces a natural transformation between the corresponding covariant Hom functors. For full proof, refer: Morphism induces natural transformation of induced covariant Hom functors

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 ==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>.