nLab enriched hom-functor

Idea

The concept of enriched hom-functor is the generalization of that of hom-functors from category theory to enriched category theory.

Definition

Definition

(enriched hom-functor)

For $\mathcal{V}$ a closed symmetric monoidal category with all limits and colimits, let $\mathcal{C}$, be a $\mathcal{V}$-enriched category. Then its $\mathcal{V}$-enriched hom-functor is the $\mathcal{V}$-enriched functor

$$\mathcal{C}(-,-) \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}$$

from the enriched product category of $\mathcal{C}$ with its enriched opposite category to $\mathcal{V}$, canonically regarded as enriched over itself, which sends pairs of objects to the corresponding hom-object in $\mathcal{C}$

$$c_1,c_2 \mapsto \mathcal{C}(c_1,c_2)$$

and which on hom-objects of $\mathcal{C}^{op} \times \mathcal{C}$ is given by the evident pre- and post-composition operation.

(e.g. Borceux 94, Prop. 6.2.7)

Related concepts

• hom-set, hom-object

• hom-functor

• enriched hom-functor, internal hom-functor

• derived hom-functor

References

• Francis Borceux, Vol 2, Prop. 6.2.7 of Handbook of Categorical Algebra, Cambridge University Press (1994)