nLab CatAdj




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

(1)Cat AdjCat Cat_{Adj} \longrightarrow Cat


[MacLane (1971), p. 103]


Relation to bifibrations


Under the Grothendieck construction, the Grothendieck fibrations which arise from pseudofunctors Cat\mathcal{B} \longrightarrow Cat that factor through Cat AdjCat_{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 adjCatCat_{adj} \to Cat is a locally full sub-2-category (this Prop.) this already means that the given pseudofunctor factors through Cat adjCat_adj, and essentially uniquely so.

See also Harpaz & Prasma (2015), Prop. 2.2.1.


See also:

Last revised on May 6, 2023 at 17:09:48. See the history of this page for a list of all contributions to it.