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 $C$, $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$, is the covariant functor defined by:
| Object functor: | \(\ds \operatorname{Hom}_{\mathbf C} \left({C, D}\right) = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) \) | ||||||||
| Morphism functor: | \(\ds \operatorname{Hom}_{\mathbf C} \left({C, f}\right): \operatorname{Hom}_{\mathbf C} \left({C, A}\right) \to \operatorname{Hom}_{\mathbf C} \left({C, B}\right), g \mapsto f \circ g \) | for $f: A \to B$ |
where $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ denotes a hom set.
Thus, the morphism functor is defined to be postcomposition.
That $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right)$ 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
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 2.7$