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Stream: learning: questions

Topic: partial inverses in $\infty$-categories

Francesco Genovese (May 11 2021 at 16:58):

Is there a way to define left/right inverses in an $\infty$-category? The $1$-categorical counterpart would be a morphism $f$ for which there exists a $g$ such that $fg=1$ (or dually).

Reid Barton (May 11 2021 at 17:43):

Yes, it's a 2-simplex whose long edge is degenerate and one of whose other edges is $f$.

Francesco Genovese (May 11 2021 at 18:25):

Reid Barton said:

Yes, it's a 2-simplex whose long edge is degenerate and one of whose other edges is $f$.

don't I need some higher data maybe?

John Baez (May 11 2021 at 18:30):

Remember, the simplicial set that looks like a 2-simplex contains degenerate simplexes of all higher dimensions. I think these provide all the higher data.

John Baez (May 11 2021 at 18:32):

(When I say "the simplicial set that looks like a 2-simplex", I really mean the "simplicial 2-simplex". There's a simplicial set called the simplicial n-simplex, which is really the representable of the object $[n] \in \Delta$. It contains a lot of higher-dimensional simplexes, but they're all degenerate.)

Francesco Genovese (May 11 2021 at 19:35):

So you mean that I can understand partial inverses actually just as partial inverses in the homotopy category?

Dmitri Pavlov (May 12 2021 at 05:57):

Although one can define higher coherence data for (homotopy coherent) inverses, it turns out to be contractible. This is valid more generally for adjunctions. See Theorem 4.4.18 in Riehl and Verity's Homotopy coherent adjunctions and the formal theory of monads.

Mike Shulman (May 12 2021 at 15:14):

@Dmitri Pavlov Do you have a reference in the case of one-sided inverses? It's hard to think offhand of what higher coherence data for such a thing would look like.

Martti Karvonen (May 12 2021 at 18:18):

@Mike Shulman
Maybe I'm misunderstanding the question (I've started learning this stuff very recently), but wouldn't the coherence data just be "all the ways of filling the relevant horn", so the claim is that the pullback of $Fun(\Delta^2,S)\to Fun(\Lambda^2_0,S)$ along $(f,id)\colon\Delta^0\to Fun(\Lambda^2_0,S)$ is contractible?

Mike Shulman (May 12 2021 at 18:35):

That's certainly not true: even in an ordinary 1-category a given map can have lots of different right or left inverses.

Martti Karvonen (May 12 2021 at 18:39):

Yeah so the claim I made is clearly false. Just to check my understanding otherwise: wouldn't this pullback still capture the "higher coherence data for a one-sided inverse"?

Martti Karvonen (May 12 2021 at 18:48):

Or is there some catch kinda like how in HoTT one needs to choose the right definition for the type "f is an equivalence"

Dmitri Pavlov (May 12 2021 at 18:59):

@Mike Shulman No, I do not have any references. In principle, this should be easy to spell out explicitly: take the cofibrant replacement of the simplicial category with two objects x, y, two morphisms f:x→y and g:y→x, and a homotopy gf→id_x. But I am not aware of any place in the literature where this is done.

Mike Shulman (May 12 2021 at 20:03):

Martti Karvonen said:

wouldn't this pullback still capture the "higher coherence data for a one-sided inverse"?

I would call it "the space of one-sided inverses". But a point of that space is still just a map equipped with one homotopy, rather than anything higher coherent.

Martti Karvonen (May 13 2021 at 00:15):

Fair enough. What's then an example of something capturing higher coherence (e.g. in the two-sided case)?

Mike Shulman (May 13 2021 at 00:20):

${\rm Fun}(NJ,S)$, where $NJ$ is the nerve of the contractible 2-object groupoid. This captures all the higher coherences for an equivalence in $S$, since $NJ$ has simplices of arbitrary dimension.

Martti Karvonen (May 13 2021 at 00:40):

So what if you replaced J with the walking arrow with a left inverse?

Mike Shulman (May 13 2021 at 00:49):

I guess that would be higher coherence data for a one-sided inverse. (-: