nLab enriched adjoint functor




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).



(enriched adjoint functors)
For ๐’ฑ\mathcal{V} a closed symmetric monoidal category with all limits and colimits (a cosmos for enrichment), let ๐’ž\mathcal{C}, ๐’Ÿ\mathcal{D} be a pair of ๐’ฑ\mathcal{V}-enriched categories.

Then an enriched adjoint pair of ๐’ฑ\mathcal{V}-enriched functors or ๐’ฑ\mathcal{V}-enriched adjunction between them

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

is a pair of ๐’ฑ\mathcal{V}-enriched functors as shown, such that the following equivalent conditions hold (Kelly, ยง1.11)

C(L(โˆ’),โˆ’)โ‰ƒD(โˆ’,R(โˆ’)). \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, ยง1.11, Borceux 94, Def. 6.7.1)

The 2-functor ๐’ฑ\mathcal{V}-Catโ†’CatCat \rightarrow {Cat} that sends an enriched category to its underlying ordinary category (i.e. the change of enrichment to the terminal category along ๐’ฑโ†’*\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.



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


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


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.


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 โ€” see the discussion there


The notion is due to

  • Max Kelly, ยง3 in: Adjunction for enriched categories, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059145]


  • 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)

Last revised on May 31, 2023 at 12:55:18. See the history of this page for a list of all contributions to it.