Monoidal adjunction

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

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 $G\circ F$.