### Context

#### Enriched category theory

enriched category theory

# Contents

## Idea

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

## Definition

###### Definition

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

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

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

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

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

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

## References

• 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)