Suppose that

(\mathcal C,\otimes,I)

and

(\mathcal D,\bullet,J)

are two

monoidal categories In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and r ...

. A monoidal adjunction between two lax monoidal functors :

(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)

and

(G,n):(\mathcal D,\bullet,J)\to(\mathcal C,\otimes,I)

is an adjunction

(F,G,\eta,\varepsilon)

between the underlying functors, such that the

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

s :

\eta:1_\Rightarrow G\circ F

and

\varepsilon:F\circ G\Rightarrow 1_{\mathcal D}

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that :

(F,m):(\mathcal C,\otimes,I)\to (\mathcal D,\bullet,J)

is a lax monoidal functor such that the underlying functor

F:\mathcal C\to\mathcal D

has a right adjoint

G:\mathcal D\to\mathcal C

. This adjunction lifts to a monoidal adjunction

(F,m)

⊣

(G,n)

if and only if the lax monoidal functor

(F,m)

is strong.

Lifting adjunctions to monoidal adjunctions

See also