Monoidal adjunction

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title: "Monoidal adjunction" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["adjoint-functors", "monoidal-categories"] topic_path: "general/adjoint-functors" source: "https://en.wikipedia.org/wiki/Monoidal_adjunction" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

A monoidal adjunction is an adjunction in mathematics between monoidal categories which respects the monoidal structure.

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

References

1. "monoidal adjunction". nlab.
2. Lindner, Harald. (1978). "Adjunctions in monoidal categories". Manuscripta Mathematica.
3. Hasegawa, Masahito. (2012-12-06). "Models of Sharing Graphs". Springer Science & Business Media.

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