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Stream: learning: questions

Topic: Functor staying within isomorphism classes on objects

Bruno Gavranović (Jan 15 2021 at 17:25):

I've got an odd question. I have an endofunctor which on objects stays within their isomorphism classes. Can it be formalized that this functor is in some sense "trivial"? Or is my intuition here wrong?

That is, $F : \mathcal{C} \to \mathcal{C}$ maps each object $A$ to an object $A'$ where $A \cong A'$. For an object $A$, call this isomorphism $i_A$.

A morphism $f : A \to B$ is mapped to $A' \xrightarrow{i_A^{-1}} A \xrightarrow{f} B \xrightarrow{i_B} B'$. It can be showed this assignment is functorial.

Now, I can't help but to feel like there's something wrong by defining a functor like this. It certainly is a functor, but it feels like it "isn't doing anything", but don't know how to state this more precisely. Can somebody see what I mean here?

Fawzi Hreiki (Jan 15 2021 at 17:32):

It’s naturally isomorphic to the identity functor.

Bruno Gavranović (Jan 15 2021 at 17:37):

Oh that was quick :grinning: I'll go and unpack this...

Amar Hadzihasanovic (Jan 15 2021 at 17:38):

I also wouldn't say that such a functor "is not doing anything" -- there are interesting functors like that, e.g. retractions of a category onto a chosen skeleton...

Bruno Gavranović (Jan 15 2021 at 17:42):

Amar Hadzihasanovic said:

I also wouldn't say that such a functor "is not doing anything" -- there are interesting functors like that, e.g. retractions of a category onto a chosen skeleton...

Maybe I should've said "from the perspective of CT it isn't doing anything". I certainly feel ill-equipped to reason about this. If some construction is naturally isomorphic to the identity, then how can any other constructions meaningfully use that construction in a different way than they would use the identity itself?

Fawzi Hreiki (Jan 15 2021 at 17:43):

You’re right. It’s not doing anything as such

Bruno Gavranović (Jan 15 2021 at 17:43):

Amar Hadzihasanovic said:

I also wouldn't say that such a functor "is not doing anything" -- there are interesting functors like that, e.g. retractions of a category onto a chosen skeleton...

Also, isn't doing skeletal stuff evil in some way?

Fawzi Hreiki (Jan 15 2021 at 17:44):

But it may be interesting that this is the case since you may not have known that a priori

Bruno Gavranović (Jan 15 2021 at 17:49):

Yeah so I've unpacked that statement. It definitely seems that it's naturally isomorphic to identity. I guess a question is: why is this bad? My attempt would be: any other construction that tries to use this functor wouldn't be able to use it in a different way that it'd use the identity functor.
But I don't know, maybe there's an even simpler way to state why this is problematic.

Fawzi Hreiki (Jan 15 2021 at 17:49):

It’s not problematic at all. Suppose you have an element of a group and prove that it’s actually the identity

Fawzi Hreiki (Jan 15 2021 at 17:50):

That may or may not be interesting depending on how that element came to your attention

Fawzi Hreiki (Jan 15 2021 at 17:50):

Same story here. If some composition of a bunch of nontrivial functors ended up being isomorphic to the identity, that would be interesting

Fawzi Hreiki (Jan 15 2021 at 17:51):

The same way if the composite of two maps is the identity, it means one of them is split epic and the other split monic

Bruno Gavranović (Jan 15 2021 at 17:54):

Ah, we came up with this functor in the context of a paper and were aiming to claim this functor is doing something useful...

Fawzi Hreiki (Jan 15 2021 at 17:54):

Well the identity is pretty useful in its own way

Bruno Gavranović (Jan 15 2021 at 17:54):

It's not a composite (which I agree would be interesting)

Amar Hadzihasanovic (Jan 15 2021 at 18:03):

I would say that the functor taking an n-dim vector space with a chosen basis to $k^n$ and a linear map to its matrix representation in this basis is very useful even though it is naturally isomorphic to the identity.

Fawzi Hreiki (Jan 15 2021 at 18:04):

Yup. The first 5 weeks of every basic linear algebra course is actually proving that this is the case.

Amar Hadzihasanovic (Jan 15 2021 at 18:07):

More in general, I think that whenever we say something like "we can assume wlog that an object is of the form x" we are implicitly using some functor naturally isomorphic to the identity...

Amar Hadzihasanovic (Jan 15 2021 at 18:08):

So it clearly it is doing something rather than nothing, otherwise we wouldn't do it :)

Matteo Capucci (he/him) (Jan 15 2021 at 18:09):

Amar Hadzihasanovic said:

I would say that the functor taking an n-dim vector space with a chosen basis to $k^n$ and a linear map to its matrix representation in this basis is very useful even though it is naturally isomorphic to the identity.

This feels a bit like cheating though since you're fixing a basis, i.e. a representation

Matteo Capucci (he/him) (Jan 15 2021 at 18:11):

I'd say that to prove something is naturally isomorphic to the identity is not necessarily a failure, it can be indeed very interesting. Though once you know this you're right in saying that any categorically mindful way of using that functor wouldn't be different from using the identity.

Bruno Gavranović (Jan 15 2021 at 18:18):

Amar Hadzihasanovic said:

So it clearly it is doing something rather than nothing, otherwise we wouldn't do it :)

Right, but this "something" is then invisible to the categorical machinery, right? None of the results/theorems from CT can be used to study it since these results/theorems don't distinguish between our functor and a functor which does nothing.

Amar Hadzihasanovic (Jan 15 2021 at 18:21):

As your cheatsheet reminds us, it can be used to prove that a left/right adjoint is fully faithful...

Amar Hadzihasanovic (Jan 15 2021 at 18:28):

@Bruno Gavranovic I don't think there's anything in the principle of equivalence which implies what you're saying. Rather, what it would say is that any "non-evil" construction that uses that functor can be transported to one that uses the identity functor by some equivalence. If the natural isomorphism is interesting/nontrivial, there's no reason why this equivalence wouldn't also be interesting/nontrivial.

Amar Hadzihasanovic (Jan 15 2021 at 18:30):

The difference would only be invisible to construction that are somehow "flattening" or "decategorifying", so that natural isomorphisms become identities.

Matteo Capucci (he/him) (Jan 16 2021 at 11:23):

I think this the same philosophy behind univalence: one should identify isomorphisms and equalities, but not in the sense of flattening, as Amar says, isomorphisms by considering them all the same, but instead by enriching your notion of equality to be isomorphism. So isomorphic things 'are equal', but the way in which they're equal is a useful, non-trivial piece of data in itself.