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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

\begin{definition}\label{EnrichedAdjunction} (enriched adjoint functors) \linebreak 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

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 $\mathcal{V}Cat$:

\begin{tikzcd}[row sep=10pt] \mathbf{D} \ar[rr, bend left=40, {\mathrm{id}}, {\ }{name=s1, swap}] \ar[r, L{description}, {\ }{name=t1, pos=.9}] \ar[from=s1, to=t1, Rightarrow, { \eta }{swap}] & \mathbf{C} \ar[rr, bend right=40, {\mathrm{id}}{swap}, {\ }{name=t2}] \ar[r, R{description}] & \mathbf{D} \ar[r, L{description}, {\ }{swap, name=s2, pos=.1}] \ar[from=s2, to=t2, Rightarrow, { \epsilon }]
& \mathbf{C} &=& \mathbf{D} \ar[r, bend left=40, {L}, {\ }{name=s3, swap}] \ar[r, bend right=40, {\mathrm{id}}{swap}, {\ }{name=t3}] \ar[from=s3, to=t3, Rightarrow, {\mathrm{id}}] & \mathbf{C} \end{tikzcd}

\begin{tikzcd}[row sep=10pt] \mathbf{C} \ar[rr, bend right=40, {\mathrm{id}}{swap}, {\ }{name=s1}] \ar[r, R{description}, {\ }{name=t1, pos=.9, swap}] \ar[from=t1, to=s1, Rightarrow, { \epsilon }{swap, pos=.3}] & \mathbf{D} \ar[rr, bend left=40, {\mathrm{id}}, {\ }{name=t2, swap}] \ar[r, L{description}] & \mathbf{C} \ar[r, R{description}, {\ }{name=s2, pos=.1}] \ar[from=t2, to=s2, Rightarrow, { \eta }{pos=.8}]
& \mathbf{D} &=& \mathbf{C} \ar[r, bend left=40, {R}, {\ }{name=s3, swap}] \ar[r, bend right=40, {R}{swap}, {\ }{name=t3}] \ar[from=s3, to=t3, Rightarrow, {\mathrm{id}}] & \mathbf{D} \end{tikzcd}

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

\end{definition} 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

\begin{example} A pair of Set-enriched adjoint functors is an ordinary pair of adjoint functors. \end{example}

\begin{example} A pair of truth values-enriched adjoint functors is equivalently known as a Galois connection. \end{example}

\begin{example} \label{StrictAdjointTwoFunctors} 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. \end{example}

\begin{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 \end{example}

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](https://doi.org/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](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), pdf]

• Francis Borceux , Def. 6.2.4 6.7.1 of of:Handbook of Categorical Algebra Vol 2, Cambridge University Press (1994)