Definition:Hom Functor
Definition
Let $\mathbf{Set}$ be the category of sets.
Let $\mathbf C$ be a locally small category.
Hom Bifunctor
The hom bifunctor on $\mathbf C$ is the covariant functor $\map {\operatorname {Hom} } {-, -} : \mathbf C^{\operatorname {op} } \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets such that:
- $(1): \quad \map {\operatorname {Hom} } {a, b}$ is the hom class
- $(2): \quad$ If $\tuple {f^{\operatorname {op} }, g}: \tuple {a, b} \to \tuple {c, d}$ is a morphism, $\map {\operatorname {Hom} } {f^{\operatorname{op} }, g}: \map {\operatorname {Hom} } {a, b} \to \map {\operatorname {Hom} } {c, d}$ is $f_* \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.
Covariant Hom Functor
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.
Contravariant Hom Functor
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: | \(\ds \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \) | ||||||||
| Morphism functor: | \(\ds \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f \) | 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.
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