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Stream: learning: questions

Topic: surprising invertibility results?

Martti Karvonen (Nov 08 2021 at 17:52):

There's results in category theory that imply that some morphism is invertible when a priori one might not have expected it. For instance, 1. Given a monoidal natural transformation $\tau$ between strong monoidal functors (or more generally, we can work with [[Frobenius monoidal functor]] s), the components of $\tau$ at a dualizable object are invertible. 2. Given a lax monoidal functor $F$ with a lax monoidal right adjoint, $F$ is in fact strong monoidal (and other instances of [[doctrinal adjunction]] ). 3. In some specific situations a given canonical morphism is invertible provided any [[non-canonical isomorphism]] exists.

Do you know other surprising invertibility results that would fit the list? I'm not trying to unify these or anything, but I'd be happy to hear about others.

Matteo Capucci (he/him) (Nov 18 2021 at 10:57):

I like this question! It might be worth to cross-post on MO

Fawzi Hreiki (Nov 18 2021 at 12:50):

An elementary by still nice example is that any full, faithful and essentially surjective functor is an equivalence. It shows that (in the presence of choice), 'functorial' does not necessarily mean 'canonical'.

Leopold Schlicht (Nov 18 2021 at 17:39):

I don't understand the last sentence. How do you define "canonical"?

Martti Karvonen (Nov 18 2021 at 18:36):

Leopold Schlicht said:

I don't understand the last sentence. How do you define "canonical"?

I didn't have a definition of "canonical" in mind, nor do I need one. However, I'll give an example of a result falling under 3 (others can be found by chasing the refs in the link): If a category has a zero object, binary products and binary coproductst, then for any pair $A,B$ of objects there is a "canonical" morphism $A+B\to A\times B$ built from identities and zero morphisms, and assembling these together over all $A,B$ results in a canonical natural transformation $(-)+(-)\to (-)\times (-)$. Now one of the theorems of the type "under suitable conditions, the canonical morphism is invertible provided any non-canonical isomorphism exists" states that this canonical natural transformation is invertible iff there is any natural isomorphism between these two functors.

Martti Karvonen (Nov 22 2021 at 17:32):

Matteo Capucci (he/him) said:

I like this question! It might be worth to cross-post on MO

Done: https://mathoverflow.net/questions/409141/surprising-invertibility-results