nLab enriched adjoint functor

Idea

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

Definition

Definition

(enriched adjoint functors)
For $\mathcal{V}$ a closed symmetric monoidal category with all limits and colimits (a cosmos for enrichment), let $\mathcal{C}$, $\mathcal{D}$ be a pair of $\mathcal{V}$-enriched categories.

Then an enriched adjoint pair of $\mathcal{V}$-enriched functors or $\mathcal{V}$-enriched adjunction between them

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

is a pair of $\mathcal{V}$-enriched functors as shown, such that the following equivalent conditions hold (Kelly, §1.11)

• there exists an adjunction between them when regarded as 1-morphisms in the 2-category $\mathcal{V}Cat$, namely $V$-enriched natural transformations

  $\eta \,\colon\, Id_{\mathbf{D}} \longrightarrow R \circ L$ (the adjunction unit) and $\epsilon \,\colon\, R \circ L \longrightarrow id_{\mathbf{C}}$ (the adjunction counit)

  such that the following zig-zag law holds in $\mathcal{V}Cat$:

• there is a $\mathcal{V}$-enriched natural hom-isomorphism between enriched hom-functors, of the form

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

If the adjunction unit and adjunction counit are (enriched natural) isomorphisms then this is called an enriched adjoint equivalence.

(e.g. Kelly, §1.11, Borceux 94, Def. 6.7.1)

The 2-functor $\mathcal{V}$-$Cat \rightarrow {Cat}$ that sends an enriched category to its underlying ordinary category (i.e. the change of enrichment to the terminal category along $\mathcal{V} \to \ast$) sends a pair of enriched adjoint functor to an ordinary pair of adjoint functors and sends an enriched adjoint equivalence to an adjoint equivalence.

Examples

Example

A pair of Set-enriched adjoint functors is an ordinary pair of adjoint functors.

Example

A pair of truth values-enriched adjoint functors is equivalently known as a Galois connection.

Example

With Cat denoting the 1-category of small strict categories equipped with its cartesian monoidal structure (via forming product categories), a pair of Cat-enriched adjoint functors is also known as a pair strict adjoint 2-functors.

Example

Assuming the axiom of choice in the underlying set theory, every Dwyer-Kan simplicial groupoid (i.e. sSet-enriched groupoid) is sSet-enriched equivalent to a disjoint union of simplicial delooping groupoids of simplicial groups — see the discussion there

Related concepts

• enriched Quillen adjunction

  • simplicial Quillen adjunction

• monoidal adjunction

References

The notion is due to

• Max Kelly, §3 in: Adjunction for enriched categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059145]

Review:

• 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, 10 (2005) 1-136 [tac:tr10, pdf]

• Francis Borceux, Def. 6.7.1 of: Handbook of Categorical Algebra Vol 2, Cambridge University Press (1994)