enriched hom-functor

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

**(enriched hom-functor)**

For $\mathcal{V}$ a closed symmetric monoidal category with all limits and colimits, let $\mathcal{C}$, be a $\mathcal{V}$-enriched categories. 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)

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

Last revised on December 19, 2019 at 13:21:59. See the history of this page for a list of all contributions to it.