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Stream: learning: questions

Topic: Left/right inverses, adjunctions and Kan extensions

Moana Jubert (Mar 05 2024 at 19:00):

A given functor may have more than one left (or right) inverse (or none, of course).

In the case of equivalences, it is always possible to get the data of an adjoint equivalence given a choice of (co)unit.

It is not true that a left (right) inverse can be turned into an adjoint in a canonical way, otherwise by uniqueness of adjoints there would be at most one left (right) inverse up to isomorphism.

But how much can we improve the situation? Specifically:

1. Assume a functor has at least one left (right) inverse. Is one of them adjoint to the given functor?
2. Suppose now that a functor has exactly one left (right) inverse (up to isomorphism). Must it be adjoint to it in some way? We know that if it has exactly one left inverse and one right inverse, then it is indeed the case as it forms an equivalence.
3. To which extent is it possible to compute left or right inverses using Kan extensions or Kan lifts? For example, if a functor $F : \mathcal{C} \to \mathcal{D}$ has exactly one left inverse $R$, is $R$ isomorphic to either the left or right Kan extension of $\mathrm{id}_{\mathcal{C}}$ along $F$?

Aaron David Fairbanks (Mar 05 2024 at 19:23):

Let $\mathcal{C}$ be the terminal category and let $\mathcal{D}$ be a nontrivial monoid regarded as a category. There is a unique functor in either direction, a section and a retraction. The functors are not adjoint, though.

Moana Jubert (Mar 05 2024 at 20:12):

Mmmmmh, indeed! I need to spend some more time thinking about it

Moana Jubert (Mar 05 2024 at 20:12):

Thank you!

Mike Shulman (Mar 05 2024 at 21:37):

Re 3, one thing you can say is that if $F$ is fully faithful, then Kan extensions along it are actual extensions: $\mathrm{Lan}_F(G) \circ F \cong G$ and dually. In particular, therefore, if $F$ is fully faithful, then $\mathrm{Lan}_F(\mathrm{Id})$ and $\mathrm{Ran}_F(\mathrm{Id})$, if they exist, are both left inverses of $F$ (but do not usually coincide).

Reid Barton (Mar 06 2024 at 17:09):

A class of rather boring examples are the discrete categories (with only identity morphisms), and functors between them, which are just functions between the underlying object sets. Then adjoints are actually two-sided inverses, because the unit and counit of the adjunctions have to be identity morphisms.