Monoidal adjunction

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

Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors

is an adjunction between the underlying functors, such that the natural transformations

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

is a lax monoidal functor such that the underlying functor has a right adjoint . This adjunction lifts to a monoidal adjunction ⊣ if and only if the lax monoidal functor is strong.

See also

• Every monoidal adjunction ⊣ defines a monoidal monad .

References