nLab CatAdj

1. Idea

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)$$Cat_{Adj} \longrightarrow Cat$$

whose

• objects are categories,

• 1-morphisms are left adjoint functors,

• 2-morphisms are natural transformations which are conjugate transformations of adjoints.

[MacLane (1971), p. 103]

2. Properties

Relation to bifibrations

Proposition 2.1. 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.

Proof. 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 Harpaz & Prasma (2015), Prop. 2.2.1.

3. Related concepts

• 2-category of adjunctions

• 2-category of model categories

• bifibration

• ∞-category of adjunctions

4. References

• Saunders MacLane, p. 102 of: Categories for the working mathematician, Graduate Text in Mathematics, Springer (1971) [doi:10.1007/978-1-4757-4721-8]

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)