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Stream: theory: category theory

Topic: pulling back adjoints

Matteo Capucci (he/him) (Mar 11 2023 at 21:06):

This Math SE post shows a counterexample to the claim pullback preserve adjunctions, but I don't understand the conceptual reason why it doesn't work in the first place.
A consequence of this is that the reindexing 2-functors of $Cat/-$ are, in fact, not 2-functorial.
So I wonder what fails and what conditions can restore this.

Taichi Uemura (Mar 12 2023 at 09:03):

The reindexing $f^{*} : Cat/D \to Cat/C$ along a functor $f : C \to D$ is a 2-functor because of the 2-dimensional universal property of pullback. So it preserves adjunctions in $Cat/D$. It need not preserve adjunctions outside $Cat/D$.

Matteo Capucci (he/him) (Mar 12 2023 at 09:44):

Ha, right

Matteo Capucci (he/him) (Mar 12 2023 at 09:44):

I confused an adjunction with domain $D$ with an adjunction in $Cat/D$, definitely not the same thing

Matteo Capucci (he/him) (Mar 12 2023 at 16:58):

The question remains, when can I pull back adjoints?

Nathanael Arkor (Mar 12 2023 at 18:40):

The pullback of a right adjoint functor between locally presentable categories, along another such functor, will also be right adjoint.

Nathanael Arkor (Mar 12 2023 at 18:41):

This follows from Bird's thesis Limits in 2-Categories of Locally Presentable Categories.

Taichi Uemura (Mar 12 2023 at 19:39):

The pullback along a fibration preserves left adjoints, found in e.g. Hermida's thesis "Fibrations, logical predicates and indeterminates".