nLab enriched hom-functor

Redirected from "enriched hom-functors".
Note: enriched hom-functor and enriched hom-functor both redirect for "enriched hom-functors".

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Idea

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 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.

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

Last revised on May 31, 2023 at 15:02:07. See the history of this page for a list of all contributions to it.