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