There are several notions of homomorphisms between pairs of adjoint functors, notably the notion of pairs of conjugate natural transformations (Def. below).
There are several layers of generality at which one may consider a notion of homomorphism between adjoint functors.
Here is a basic but important notion:
(conjugate transformations of adjoints)
Given a pair of pairs of adjoint functors between the same categories
then a pair of natural transformations between the adjoints of the same chirality, of this form
is called conjugate for [MacLane (1971), ยงIV.7 (5)] or a pseudo-transformation of [Harpaz & Prasma (2015), Sec. 2.2] the given adjunctions if they make the following diagram of natural transformations between hom-sets commute:
where the horizontal maps are the given hom-isomorphisms (see there).
Such pairs of conjugate transformation compose via composition of their component natural transformations (cf. functor category) to yields a category
whose
objects are the left adjoint functors $\mathcal{C} \to \mathcal{D}$
morphisms are the conjugate transformations between these
composition is the composition of component natural transformations.
Given a pair of pairs of adjunctions as in (1) and given (just) $\lambda$ as in (2) then there exists a unique $\rho$ as in (2) such that the conjugacy condition (3) holds.
In other words, the evident forgetful functor from $Cat_{adj}(\mathcal{C},\,\mathcal{D})$ (4) to the hom-category $\Cat(\mathcal{C},\,\mathcal{D})$ (i.e. the functor category) is a fully faithful functor exhibiting a full subcategory-inclusion:
In fact:
(conjugate pairs are mates)
The conjugacy condition (3) means equivalently that $\lambda$ and $\rho$ are mates in the sense of 2-category theory.
Moreover:
Conjugacy of transformations is compatible with horizontal composition $(-)\cdot(-)$ of natural transformations as 2-morphisms in Cat (โwhiskeringโ), so that (5) extends to a horizontal composition-functor:
Therefore, from Prop. and Prop. we have:
The (very large) wide and locally full sub-2-category $Cat_{adj}$ of Cat
whose
objects are categories,
2-morphisms are natural transformations which are conjugate in the sense of Def. .
Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{adj}$ (6) are equivalently the bifibrations.
A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. Jacobs (1998), Lem. 9.1.2). By the fact that $Cat_{adj} \to Cat$ is a locally full sub-2-category (Prop. ) this already means that the given pseudofunctor factors through $Cat_adj$, and essentially uniquely so.
See also:
Yonatan Harpaz, Matan Prasma, Section 2.2. of: The Grothendieck construction for model categories, Advances in Mathematics 281 (2015) 1306-1363 [arXiv:1404.1852, 10.1016/j.aim.2015.03.031]
(in the context of model structures on Grothendieck constructions)
Last revised on September 27, 2023 at 20:09:19. See the history of this page for a list of all contributions to it.