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Monoidal adjunction

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

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

( F , m ) : ( C , ⊗ , I ) → ( D , ∙ , J ) {\displaystyle (F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)} and ( G , n ) : ( D , ∙ , J ) → ( C , ⊗ , I ) {\displaystyle (G,n):({\mathcal {D}},\bullet ,J)\to ({\mathcal {C}},\otimes ,I)}

is an adjunction ( F , G , η , ε ) {\displaystyle (F,G,\eta ,\varepsilon )} between the underlying functors, such that the natural transformations

η : 1 C ⇒ G ∘ F {\displaystyle \eta :1_{\mathcal {C}}\Rightarrow G\circ F} and ε : F ∘ G ⇒ 1 D {\displaystyle \varepsilon :F\circ G\Rightarrow 1_{\mathcal {D}}}

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

( F , m ) : ( C , ⊗ , I ) → ( D , ∙ , J ) {\displaystyle (F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)}

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

See also

  • Every monoidal adjunction ( F , m ) {\displaystyle (F,m)} ⊣ ( G , n ) {\displaystyle (G,n)} defines a monoidal monad G ∘ F {\displaystyle G\circ F} .

References

  1. "monoidal adjunction". nlab. Retrieved 2024-12-23.
  2. Lindner, Harald (1978). "Adjunctions in monoidal categories". Manuscripta Mathematica. 26 (1–2): 123–139. doi:10.1007/BF01167969. ISSN 0025-2611.
  3. Hasegawa, Masahito (2012-12-06). Models of Sharing Graphs. London: Springer Science & Business Media. p. 64. ISBN 978-1-4471-0865-8.
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