category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A monoidal adjunction is an adjunction between monoidal categories which respects the monoidal structure.
Since there are several types of monoidal functors (lax, colax, and strong) there are several types of “adjunctions between monoidal categories which respect the monoidal structure.” Namely, we could have:
An adjunction in the 2-category MonCat of monoidal categories and strong monoidal functors. In this case both the left and right adjoint are strong.
We call this a strong monoidal adjunction.
An adjunction in the 2-category MonCat${}_\ell$ of monoidal categories and lax monoidal functors. In this case the right adjoint is lax, while the left adjoint is necessarily strong (by doctrinal adjunction; see here).
In fact, since the right adjoint of an oplax monoidal functor is necessarily a lax monoidal functor (this prop.), it is sufficient to demand that $L$ be strong monoidal.
This version, which is one of the most frequently occurring, is often called simply a monoidal adjunction.
The dual: an adjunction in the 2-category $MonCat_c$ of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong. One might call this an opmonoidal adjunction.
A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are mates under the adjunction. This is a conjunction in the double category of monoidal categories and lax and colax monoidal functors, so we may call it a monoidal conjunction or a lax/colax monoidal adjunction. By doctrinal adjunction, given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.
As mentioned above, the nature of monoidal adjunctions follows as a special case from generalities of doctrinal adjunctions. For the record, here is an explicit discussion:
Let
be a pair of adjoint functors between monoidal categories, such that the left adjoint $L$ is a strong monoidal functor by natural isomorphisms
and
Then
the right adjoint $R$ becomes a lax monoidal functor via natural morphisms
and
where $\eta$ denotes the adjunction unit and $\epsilon$ denotes the adjunction counit, as usual.
For any object $A \in \mathcal{D}$ carrying the structure of a monoid object $(A, \mu_A, e_A)$, then
the unit of the adjunction $\eta(A) \;\colon\; A \longrightarrow R L(A)$ is a monoid homomorphism with respect to the canonically induced monoid structure on $R L(A)$ (this prop.) given by
and
similarly for the counit of the adjunction.
The first statement is discussed at oplax monoidal functor.
For the second statement, we need first need to check that the following square commutes:
Now by definition, the composite of the top and right morphism here is the total diagonal composite in the following diagram:
Here the top sqaures commute by naturality of $\eta$ and $\mu_L$, and top right diagonal morphism is the identity morphism, as shown, by the zig-zag identity for the adjunction $(L \dashv R)$. Therfore $R(\mu_L^{-1})$ cancels agains $R(\mu_L)$. so that the composite morphism in question becomes just $A \otimes A \overset{\eta(A \otimes A)}{\longrightarrow} R L(A \otimes A) \overset{ R L(\mu_A) }{\longrightarrow} R L(A)$. Again by the naturality of the adjunction unit $\eta$
this equals $\eta(A) \circ \mu_A$, as required.
Finally we need to check that the following diagram commutes:
Now unwinding the above definitions of $e_{R L}(A)$ in terms of the definition of $e_R$ we find that
Here the two morphisms in the middle cancel, so that we are left just with
That this equals $\eta(A)\circ e_A$, as required, follows by the naturality of $\eta$:
The argument for the homomorphism property of the counit should be formally dual to the above.
(stabilization in stable homotopy theory)
The stabilization adjunction
between the classical homotopy categories $Ho(Spaces)$ and $Ho(Spaces^{\ast/})$ of (pointed) topological spaces and the stable homotopy category $Ho(Spectra)$ is a monoidal adjunction, since the left adjoint $\Sigma^\infty(-)_+$ (forming the suspension spectrum of a space after freely adjoining a basepoint) is strong monoidal with respect to forming product topological spaces and forming smash product of spectra, respectively.
In fact this is the derived functors of what is even a monoidal Quillen adjunction between the classical model structure on topological spaces and the stable model structure on orthogonal spectra (this cor.), which implies (strong) monoidality of the derived functors on homotopy categories (this prop.).
In detail, let
be the Quillen adjunction on orthogonal spectra (here). The left adjoint $L$ is a strong monoidal functor, and hence so is its derived functor $\Sigma^\infty(-)_+ \colon Ho(Top) \to Ho(Spectra)$ (by this prop.).
We want to see that the structure of a lax monoidal functor which is induced on the derived right adjoint $\Omega^\infty(-) \colon Ho(Top) \to Ho(Spectra)$ via prop. is the expected one, given on Omega-spectra $X$ and $Y$ by
To see this, observe that if $X$ and $Y$ are CW-Omega-spectra and hence cofibrant and fibrant in $\mathbb{S}_{orth}Mod_{stable}$ then the derived lax monoidal structure is given by the total bottom composite in the following diagram
where we write for brevity $(L \dashv R) \coloneqq (\Sigma^\infty_{orth} \dashv \Omega^\infty_{orth})$, and where $j \colon id \longrightarrow P$ denotes functorial fibrant replacement (which exists since the small object argument applies in $\mathbb{S}_{orth}Mod_{stable}$). By functoriality of the replacement, all the squares commute, so that the derived lax monoidal structure on CW-Omega spectra is seen to be equivalently the underived one.
But that underived top horizontal composite is manifestly just the canonical isomorphism $X_0 \wedge Y_0 \simeq (X \wedge Y)_0$ (since $R$ simply picks the component space in degree-0 and $L$ preserves the component space in degree 0).
(exponential modality in linear type theory)
In linear type theory (see there for more) the exponential modality $!$ may have categorical semantics as the comonad induced by a monoidal adjunction.