@DenisNardin I think my attempts at sending messages from last night show replying to people doesn't always tag them, so I don't know if you'll see this, but will do.

Thanks chaps this is helpful. @HarryGindi , I've already hung this beauty on my refrigerator! Sorry to press again, but I can't move forward until I untangle something here. First, I have been considering sAlg_k as an infinity category arising as the nerve of the ordinary category of sAlg_k, so I took simplicies in sAlg_k to be n-fold compositions.

I'm happy to view say a simplex U -> U -> V as a triangle, but it seems the simplexes are not commutative but only commutative up to homotopy is your point, however I don't know how to unwind that. For example, if I wanted to build a map of the form Hom_{sAlg_k}^R(U,V) \to Hom_{sAlg_k}^R(U,W) what does this look like? Say in fixed degree, it must take an n-simplex in Hom_{sAlg_k}^R(U,V) to one in Hom_{sAlg_k}^R(U,W). Can I represent an arbitrary element of Hom_{sAlg_k}^R(U,V)_n as a diagram

U -> U -> V (considered as a triangle with edge U->U identity, and two maps U->V considered homotopic)? To produce its image, I just need to specify maps two maps U -> W, but what do I look for to ensure these are homotopic? If there is a map say V -> W, does post-compose all simplicies with this map trivially induce a map on these simplicial sets?

Despite the excellent descriptions above, thanks again @DenisNardin and @HarryGindi, I fear I'm still lacking in something fundamental here, so any pointing to where my misconceptions are, no matter how obvious they might be, would be very helpful

actually, later I plan to work in dAff, so I guess sAlg_k should be taken as a localization of the usual model structure

This I guess changes what the simplicies really are?

(above n = 1, higher n's will be clear I think once I see what I'm doing there. Also @Harry

also @HarryGindi I know these have to compatible with various face/degeneracy, i.e., its a map of simplicial sets. I'm trying to get a handle on the level wise maps)

@lemiller If you're taking the 1-category sAlg_k, then you're right: you're only allowing constant homotopies so the resulting simplicial set is just the Hom set of maps (seen as a constant simplicial set). But if, as I imagine, you're more interested in the ∞-category dAlg_k obtained by inverting the weak equivalences, the precise simplicial set you get starts to depend on the exact construction of dAff_k you're considering (since we're after all discussing a model)

If you specify one further I'll be happy to help you work out exactly which sset you end up with

@DenisNardin, you are absolutely correct! I guess the standard one would be analgous to simplicial commutative rings (as sAlg_Z) to take fibrations the underlying fibrations of underlying simplicial sets, and weak equivalences the same (determining cofibrations from this)

but I'm guessing that's the one you had in mind when originally answering my question?

@lemiller That's specifying a model category. Sorry to be pedantic, but how are you getting an ∞-category out of that? As I said we're trying to figure out details of a model, and so we need to be precise about this stuff...

To be less obnoxious, since this is a simplicial model category, are you just taking the simplicial nerve of bifibrant objects?

sorry, I think one puts that model structure on the nerve of sAlg_k, then localizing that infinity category at weak equivalences, at least that is what I'm reading is done say by Toen. Is that on the mark?

In HTT I've seen Lurie go back and forth between different formulations of infinity categories, but I'm not sure if that's the type of manipulation that you're describing of bifibrant objects (not sure I've seen a definition of those)

perhaps if I'm localizing I don't need to first take a nerve?

(of course thanks a bunch in advance!)

@lemiller That specifies an ∞-category up to equivalence, which is of course all you normally need, by giving a universal property. But let's be more concrete for the sake of getting a concrete model on our hands

Ah I see, so there is more input needed than this to determine what the simplicies are in the mapping space?

Yeah, this just determines the homotopy type of the mapping space (which, again, is normally all you need)

I cannot remember the precise reference right now (it boils down to some theorems by Dwyer and Kan), but if you have a simplicial model category M you can get a quasicategory representing M[w^{-1}] by taking the simplicial nerve of the simplicial subcategory of M spanned by bifibrant objects

By which I mean objects such that $\varnothing \to x$ is a cofibration and $x\to \ast$ is a fibration

Then in this particular case (the simplicial nerve of a category) the mapping space is represented exactly by the simplicial set Hom(x,y) (again this in HTT, I can hunt the precise references later if you want). Note that the condition that x,y are bifibrant forces this sset to be a Kan complex

This is a different model than the one we were talking about, but it represents the same homotopy type

That gives me a place to start, thanks a bunch! Here, I'm taking it that if I want to apply Hom(x,y) in general, I do some bifibrant replacement of both arguments, and then n-simplicies are just the level wise ordinary homs compatibly chosen with face/degeneracy maps?

(or rather I'm asking if that interpretation of hom(x,y) is what you intended)

I don't know what you mean with "levelwise homs".. They are the simplicial sets given by the enrichment of sAlg_k in ssets, which comes by seeing sAlg_k as $\operatorname{Fun}^\times(Free_k^{op}, sSet)$

Here Free_k is the category of free finitely generated k-algebras

This is described in HTT.5.5.9, if you need a reference, although of course it's much older

I think I'm taking the simplistic naive assumption that as x and y are simplicial k-algebras, but all the more simplicial sets, an element of hom(x,y) must be some homomorphism of simplicial sets (so maps on simplicies x_n -> y_n satisfying commutativity with face/degeneracy; this is what I was thinking with level-wise homs), which perhaps is erroneous and creating unnecessary confusion. Is the simplicial nerve construction spanned by bifibrant objects discussed there?

(I don't see bifibrant, is this perhaps the same as fibrant-cofibrant?)

@lemiller Those are exactly the 0-simplices, but you also need to define the higher simplices :)

I could email with more details if you feel that is more efficient or helps cut down on the back/forth.

I do too, which works great for this format, but on longer discussions, I find these small textboxes a bit limiting. I'll prep an email so the chat isn't flooded with this, but thanks a bunch!

lemiller I missed the whole discussion, but salg_k has a universal property as the free cocompletion under sifted colimits of finite polynomial rings over k

(just woke up)

in practice you can think of this as a way of noticing that all cofibrant simplicial algebras are levelwise free

@HarryGindi Be careful about the notation: they're using sAlg_k for the 1-category of simplicial algebras and dAlg_k for what you get after inverting w.e.

oh no, Arpon and Akhil use dAlg for their new version of ' nonconnective simplicial rings'

AAlg, animated algebras

I used this terminology in the paper I'm writing =]

(following Clausen and Scholze =]. I didn't make it up.)

@HarryGindi thanks, what work is that of Apron/Akhil? @DenisNardin again thanks, this has all been incredibly helpful. I won't hesitate to continue the discussion as questions arise. I hope only that my own naive silly questions inspire bravery in others!

Look up Arpon's latest paper on arXiv I think

Akhil hasn't put anything up yet about it

but Arpon mentions in the paper that the definition either comes from upcoming work by Akhil or was something they came up with together

I didn't read it in detail but it's the paper involving filtered monads

@HarryGindi Awesome, thanks!

stop pingimg meeeeee

Oh, sorry. I thought that was custom. I'll do this less often

no it makes my computer beep every time

I'm still here =]

ah, you know you can mute that if you want to. The speaker in the upper right is clickable.

i like it when it isn't abused!!!!

Also, given Arend and Milena my best when you see them next :)

=]

who are you?

Oh, sorry, my name is Lance

the handle gives the rest away

I can't very well sat "MO user lemiller says hi"

they'll go into hiding lol

ok ttyl buddy

Is there some sense in which $Ω^∞S^{tC_2}=Ω^∞S^\wedge_2$ is an algebraic approximation for $BGL_1(S)$? For context I'm trying to get a conceptual understanding of how a chain bundle is an approximation to a stable spherical fibration

S here is the sphere spectrum?

@lemiller Yeah

Actually traditionally a chain bundle is a map Σ^∞_+X→(HR)^{tC_2}. but I don't see any reason why the analogy should work only for ordinary rings, and in fact my intuition tells me it should work better for the sphere spectrum

What is the automorphism group of the little n-cubes operad? (I feel like I ought to know this, but apparently not ...)

I want to say its the automorphisms of $S^{n-1}$ ...

more generally, what's $\mathrm{Map}(\mathbb{E}_m, \mathbb{E}_n)$?

When n-m > 2, this is the space whose (m+1)-fold loopspace is the space of higher dimensional long knots, I think. This was Dwyer-Hess and then Boavida de Brito-Weiss for the general case.

I have a memory that Dwyer and Hess thank you (Charles) for some instrumental idea, so I'm guessing you knew that.

Some expert should correct me if I'm talking rot. I'm operating from memory.

I think your memory is correct (e.g. arxiv.org/pdf/1502.01640.pdf). I think not much is known about these things, and I think some folks think the answer for automorphisms is not Top(n), for example. It's also worth pointing out that if you start completing these operads, or looking rationally, you get all sorts of interesting things...