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

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

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

c 1,c 2𝒞(c 1,c 2) c_1,c_2 \mapsto \mathcal{C}(c_1,c_2)

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

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


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