Definition:Covariant Hom Functor

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Definition

Let $\mathbf{Set}$ be the category of sets.

Let $\mathbf C$ be a locally small category.

Let $C \in \mathbf C_0$ be an object of $\mathbf C$.


The covariant hom functor based at $C$, $\map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}: \mathbf C \to \mathbf{Set}$, is the covariant functor defined by:

Object functor:    \(\ds \map {\operatorname{Hom}_{\mathbf C} } {C, D} = \map {\operatorname{Hom}_{\mathbf C} } {C, D} \)      
Morphism functor:    \(\ds \map {\operatorname{Hom}_{\mathbf C} } {C, f}: \map {\operatorname{Hom}_{\mathbf C} } {C, A} \to \map {\operatorname{Hom}_{\mathbf C} } {C, B}, g \mapsto f \circ g \)      for $f: A \to B$

where $\map {\operatorname{Hom}_{\mathbf C} } {C, D}$ denotes a hom set.


That $\map {\operatorname{Hom}_{\mathbf C} } {C, \cdot}$ is a functor is shown on Covariant Hom Functor is Functor.


Also known as

Some sources call a hom functor a representable functor.



Also denoted as

All notations for hom classes can be seen for hom functors too.



A covariant hom functor can also be denoted $h^x$; see the Yoneda embedding.


Also see


Sources