Monoidal adjunction

In mathematics, a monoidal adjunction is an adjunction between monoidal categories which respects their monoidal structures.

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

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

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

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

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

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

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

See also

• Every monoidal adjunction $(F,m)$⊣$(G,n)$ defines a monoidal monad $G\circ F$.