Definition:Covariant Hom Functor

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.

This article is complete as far as it goes, but it could do with expansion. In particular: And this is inconvenient because of concept overloading, but we haven't covered the other type of representable functor You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also denoted as

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

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

Also see

• Contravariant Hom Functor

Sources

• 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.7$