Definition:Contravariant 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 contravariant hom functor based at $C$:
- $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$
is the covariant functor defined by:
| Object functor: | \(\displaystyle \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \) | |||||||
| Morphism functor: | \(\displaystyle \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f \) | for $f: A \to B$ |
where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set.
Thus, the morphism functor is defined to be precomposition.
Also known as
Some sources call a hom functor a representable functor.
And this is inconvenient because of concept overloading, but we haven't covered the other type of representable functor
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Also denoted as
All notations for hom classes can be seen for hom functors too.
The above should be explicated
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A contravariant hom functor can also be denoted $h_x$; see the Yoneda embedding.
Also see
- Contravariant Hom Functor is Functor, where it is shown that $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right)$ is a functor.
Sources
- 2010: Steve Awodey: Category Theory ... (previous) ... (next): $\S 5.5$
