Hi Jesparza, welcome. Your question seems totally suitable for the chatroom, but unfortunately I don't work in this direction so I cannot help you.

@jesparza In my understanding, no significant problem in number theory was solved by tmf. Sometimes, one gets a new perspective though. See for example Theorem 5.10 in Hopkins's ICM talk: arxiv.org/pdf/math/0212397.pdf -- A new topological proof was found of a number-theoretic result. After the fact, the proof was translated back to a new non-topological proof.

what happens even more often is that tmf brings some questions to attention that number theorists apparently found not interesting enough to solve so far. An example is the computation of H^i(M_{ell}; \omega^{\otimes j}) - a problem in arithmetic geometry, but only solved by algebraic topologists because this is the input of a spectral sequence computing the homotopy groups of TMF. I've also written two arithmetic articles inspired by questions coming from TMF.

In HTT 4.2.3.15 describes a functorial construction which to every simplicial set assigns a poset with a cofinal functor into it. He then says that if one drops the functoriality one can make sure that if the original simplicial set is finite the poset can also be arranged to be finite. I was wondering if this can be made functorial at least in the infinity cateegorical setting.

That is: Is there a functor from the category of finite $\infty$-categories to the category of finite posets and a natural transformation from the composition of it with the fully faithful inclusion (of finite posets into finite infinity-categories) to the identity which is cofinal?

@SaalHardali I think this is some kind of minimal fibration argument, so I doubt it.

Yeah I briefly looked at it and that's what it loked like to me

Still the statements aren't really comparable because I'm asking for an infinity-functor not a strict functor on the 1-cateogories

I think at the level of an ∞-functor it's already functorial then, right?

Not sure what you mean

like, in the appropriate ∞-category where this takes place, the poset and the finite poset should be homotopically indistinguishable somehow, but I dunno. I'll think about it and get back to you.

Aha, thanks!

I don't know exactly how to state it, but I think it's going to be like this takes place in some category of left or right fibrations over the simplicial set you're working with

and the existence of such a finite poset will be a proof that the functorial guy is homotopically finite in that appropriate category, but this is all just bs til I work it out

So just to make sure were on the same boat, i'm asking if every finite infinity category admits an (infinity) functorial cofinal functor from a finite poset

(my phrasing was a bit awkward before)

My answer that I'll try to work out later is I think so, but the meaning of 'finite poset' has to change to 'a poset weakly equivalent in some fibered sense to a finite poset'

hmm i'll think about that too, thanks!

@ThomasRot @LennartMeier thanks! your answers were helpful, I'll look more into the theorem into Hopkins's talk! :)

does Peter Haine come through here? I had some simple questions regarding his exodromy paper with Glasman and Barwick