Monoidal adjunction

A monoidal adjunction is an adjunction in mathematics between monoidal categories which respects the monoidal structure.{{cite web | title=monoidal adjunction | url=https://ncatlab.org/nlab/show/monoidal+adjunction |publisher=nlab | access-date=2024-12-23}}{{cite journal | last=Lindner | first=Harald | title=Adjunctions in monoidal categories | journal=Manuscripta Mathematica | volume=26 | issue=1-2 | date=1978 | issn=0025-2611 | doi=10.1007/BF01167969 | pages=123–139}}{{cite book | last=Hasegawa | first=Masahito | title=Models of Sharing Graphs | publisher=Springer Science & Business Media | publication-place=London | date=2012-12-06 | isbn=978-1-4471-0865-8 | page=64}}

Suppose that $(\mathcal C,\otimes,I)$ and $(\mathcal D,\bullet,J)$ are two monoidal categories. A monoidal adjunction between two lax monoidal functors

: $(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)$ and $(G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I)$

is an adjunction $(F,G,\eta,\varepsilon)$ between the underlying functors, such that the natural transformations

: $\eta:1_{\mathcal C}\Rightarrow G\circ F$ and $\varepsilon:F\circ G\Rightarrow 1_{\mathcal D}$

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

: $(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)$

is a lax monoidal functor such that the underlying functor $F:\mathcal C\to\mathcal D$ has a right adjoint $G:\mathcal D\to\mathcal C$ . This adjunction lifts to a monoidal adjunction $(F,m)$ ⊣ $(G,n)$ if and only if the lax monoidal functor $(F,m)$ is strong.

References

Category:Adjoint functors

Category:Monoidal categories

Monoidal adjunction

Lifting adjunctions to monoidal adjunctions

See also

References