There are different versions of what one may mean by a 2-category of categories with adjoint functors as 1-morphisms, namely depending on which notion of transformation of adjoints one considers.
For example, the (very large) wide locally full sub-2-category of Cat
whose
objects are categories,
2-morphisms are natural transformations which are conjugate transformations of adjoints.
Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors $\mathcal{B} \longrightarrow Cat$ that factor through $Cat_{Adj}$ (1) 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 (this 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 May 6, 2023 at 17:09:48. See the history of this page for a list of all contributions to it.