4:24 AM
@ReubenStern Thanks Reuben!

5 hours later…
9:20 AM
@DenisNardin I am in the process of elaborating in a TeX file.

2 hours later…
11:19 AM
Does anyone know a reference (not a proof) for the following fact?
A small ∞-category is stable iff it has all finite limits and finite colimits and admits a conservative exact functor to the ∞-category of spectra.

11:49 AM
(I ask because I just invoked this fact in a MO answer: mathoverflow.net/a/374704/57405)

@AlexanderCampbell I don't know a reference but it's literally half a line using HA.1.4.2.27, so I would use that as a reference

@DenisNardin I don't see how to prove the 'only if' direction (which is the interesting direction) from that reference. (I give a proof in the linked answer.)

Ahh I missed the only if. Sorry for the noise

Not at all :)
@ScottBalchin I see you're giving a lecture series on model categories next month! sarah-whitehouse.staff.shef.ac.uk/ttt/…
2

@AlexanderCampbell Yes, the LMS decided it would be a good idea to run some introductory style things for incoming PhD students :) Of course it gives me a good chance to talk about some of the more exotic examples from the book

12:03 PM
I look forward to watching it. It'll be late at night in my time zone, so I'm glad they'll be recorded.

4 hours later…
4:03 PM
@DanielBruegmann i'd be interested to see this, too! that is a cool fact.
@ScottBalchin congrats, that looks like fun! what book is @AlexanderCampbell referring to?
@TimCampion in re your question, i believe i've seen a characterization of stable ∞-categories $C$ as follows: they admit finite limits and colimits, and moreover for any functor $I \times J \to C$ where $I$ and $J$ are finite ∞-categories, the operations of (fiberwise) colimit over I and (fiberwise) limit over J commute. that is, LKan followed by RKan in the composite $I \times J \to J \to pt$ is equivalent to RKan followed by LKan in the composite $I \times J \to I \to pt$.
i think i learned this from moritz groth
maybe it's due to him, as well
here's a question i'm scared to know the answer to: does anybody know whether NSF grant funding this year will be the same as in past years? (or will it be reduced due to covid for some reason or another?) the program officers weren't willing to say anything definitive, but i figure there could still be people out there who have a better guess than i do (based on an undertanding of how the system works), or even inside info.

4:29 PM
@DanielBruegmann I'd love to see it too (if that wasn't clear :) )

4:40 PM
@AaronMazel-Gee Moritz Rahn* (these days) proves at least in derivators: a derivator (admitting finite limits and colimits) is stable if and only if all finite limits commute with finite colimits. (Co)limits of quasicategories are the same in their associated derivator

5:22 PM
Hi friends, I have another most likely dumb question about a line in HA (also do let me know if these aren't appropriate for this channel). On page 1142 of HA (latest version), above remark 14.1.1.5, he talks about the homotopy fiber of a map Map(A\otimes B, k) -> Map(A, k) \times Map (B, k). My question is, since the tensor product is the coproduct in Alg_k, isn't that map just an equivalence and hence the fiber is contractible? What am I missing?
Ah wait I'm dumb, Alg_k are E_1 not E_infty algebras, ignore the above

Ok I can't even find a version of HA with a chapter 14 so I'm glad you figured it out

Ah I meant SAG, sorry, thx

1 hour later…
6:43 PM
If I want to have a relative category, or category with w.e.'s, but it's simplicially enriched, are there requirements that need to be checked for these two things to be "compatible?"

@JonathanBeardsley In order to do what? A simplicially enriched category presents an oo-category. The week equivalences then give you some maps (just need homotopy classes thereof even), and you can ask for the universal category inverting those maps.

Maybe you should make sure that degenerations of w.e. are still w.e.?

I suppose my initial thought was just about the difference between the simplicial localization of the category with w.e.'s and the simplicially enriched category. Is there any reason to believe these things coincide, for instance?
(I think this is the case for a simplicially enriched model category, right?)

It is clear that you need some assumptions for that (consider the case where only isomorphisms are weak equivalences), but I don't know what they are.

I agree.
Here's another question: if I have a Kan complex X and I just restrict it to the full subcategory of Δ^{op} spanned by [0] and [1], can I call the resulting thing the fundamental groupoid of X? Is there any obvious flaw in this?

6:59 PM
I don't think you can recover pi_1 of your space from this, can you?

@JonathanBeardsley I'm not sure what you mean, but whatever you mean should probably include the value at [2] to know anything about composition

Right, yeah, the composition is going to get lost.

I think if you then right Kan extend back up then this would be the (nerve of the) fundamental groupoid

7:38 PM
If you have a Kan complex, you can recover its fundamental groupoid from the restriction to Δ≤2
iirc
this is because for quasicategories, the homotopy category can be computed via a coskeleton
for Kan complexes, there's also a general statement that the n+1-coskeleton computes the nth truncation
I might be indexing off-by-1

7:53 PM
Okay right, yeah, great! You extend back to sSet and you get the fundamental groupoid. I guess I was sort of implicitly thinking that but not, you know, saying it, which is a problem I have.

8:21 PM
It's true of any simplicial set that its fundamental groupoid can be recovered from its 2-skeleton.

8:42 PM
@AlexanderCampbell you mean the fundamental groupoid of the realization?

If you like, but I really mean the left adjoint to the nerve functor from groupoids to simplicial sets.

9:15 PM
@IanColey ah, right! thanks for that reminder. (including the reminder that moritz changed his last name.)
@JonathanBeardsley yes, this is true for simplicial model categories. this is a rather nontrivial theorem of dwyer--kan. one sort of compatibility you can ask for is that the weak equivalences become equivalences in the simplicial localization. equivalently, for any weak equivalence $X \to Y$, the map $hom(-,X) \to hom(-,Y)$ is a componentwise weak equivalence of simplicial sets.
(but of course, that's still very far from saying that the localization at the weak equivalences coincides with the given simplicial enrichment.)

2 hours later…
11:35 PM
i've got a relative $\infty$-category $(C,W)$, and i know that the localization $C[W^{-1}]$ is in fact an ordinary category. i'd like to conclude that $C$ itself is an ordinary category. is this true? it feel to me like it's probably false as stated.
in my actual situation of interest, $C$ just has one equivalence class of object. so i'm talking about an $\infty$-monoid $M$ whose $\infty$-group completion $G$ is in fact a discrete group. and a special feature of my situation is that the map $M \to G$ is injective on $\pi_0$. (i believe this is not automatic?) but so, i'm asking whether i can conclude that $M$ is in fact a discrete monoid.