nLab enriched adjunction




The concept of enriched adjunction is the generalization of that of adjoint functors (adjunctions in Cat) from category theory to enriched category theory.



(enriched adjunction)

For 𝒱\mathcal{V} a closed symmetric monoidal category with all limits and colimits, let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories. Then an adjoint pair of 𝒱\mathcal{V}-enriched functors or enriched adjunction

𝒞RL𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

is a pair of 𝒱\mathcal{V}-enriched functors, as shown, such that there is a 𝒱\mathcal{V}-enriched natural isomorphism between enriched hom-functors of the form

𝒞(L(),)𝒟(,R()). \mathcal{C}(L(-),-) \;\simeq\; \mathcal{D}(-,R(-)) \,.

(e.g. Borceux 94, Def. 6.7.1)

The 2-functor 𝒱\mathcal{V}-CatCatCat \rightarrow {Cat} that sends an enriched category to its underlying ordinary category sends an enriched adjunction to an ordinary adjunction.



  • Max Kelly, section 1.11 of Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64, 1982, Republished in: Reprints in Theory and Applications of Categories, No. 10 (2005) pp. 1-136 (pdf)

  • Francis Borceux, Vol 2, def. 6.2.4 of Handbook of Categorical Algebra, Cambridge University Press (1994)

Last revised on October 11, 2021 at 08:56:52. See the history of this page for a list of all contributions to it.