So, what's the... nerve of the nerve? There's a functor $N: sCat\to sSet$. We know that this induces a functor of quasicategories $Cat_\infty\to Cat_\infty$, and I think this should be homotopic to the identity, but I'm not sure...

@JonathanBeardsley this follows from the quillen equivalence between sCat and the sSet (w/ Joyal model structure) right? since your functor is implicitly compose sSet->sCat->sSet

Hrm, I'm not sure about being implicitly composed. I feel like it might follow from the fact that there are only two automorphisms of $\Cat_\infty$ up to homotopy?

I mean, I'm looking at a 1-simplex in the large quasicategory $sSET$ I guess.

another question: is it just me or does the entirety of Lurie's discussion of excellent model categories in HTT A.3.2. seem to rely on an unstated assumption that the monoidal structure is Cartesian? let C be a S-enriched category and K an object in S (S an excellent model category). then he defines C^K as the category w/ same objects and where Hom_{C^K}(X,Y) is Hom(X,Y)^K - as far as I can tell, there is no clear way to define composition or identity in this category without assuming

that the monoidal structure is Cartesian. e.g. identity is a map 1->Hom(X,Y)^K, or a map K->Hom(X,Y), and I see no reason for this to exist unless your unit is a final object

@JonathanBeardsley whats your definition of the functor QCat->QCat then? I thought based on what you were saying you were going QCat->sCat->QCat - are you doing something else?

@JonathanBeardsley Oh boy, that's a nasty question, isn't it?

@JonathanBeardsley Representing Cat_infty by taking the relative nerve of sCat is some crazy thing

but it's highly indirect

@HarryGindi yeah that's what I think too. It's an invertible self map of Cat_∞ which isn't op. I'm actually okay with this, despite it being indirect.

@dhy right, so not the composition, but like, noticing that the nerve functor induces an equivalence on the underlying quasicategories of sCat and sSet

but how are you passing from sSet to sCat if not through the inverse of nerve then? as in how does sCat enter the picture other than through taking the inverse of the equivalence you mention

9 hours left to vote

:-)

@JonathanBeardsley I'm a bit confused: isn't that simply the identity? (I mean, isn't your nerve functor one of the pieces of the Quillen equivalence between the Berger and the Joyal model structures?

@DenisNardin @dhy yeah so this is the right adjoint of a Quillen equivalence, and by a few different theorems this induces an equivalence from Cat_∞ to itself (in the large quasicategory of small quasicategories). But it doesn't seem that this just automatically makes it (equivalent to) the identity, unless I'm missing something

Actually, how are you defining Cat_∞?

And how are you identifying sSet[w^{-1}] and sCat[DK^{-1}] with Cat_∞?

'cause the identification I use uses N, so it is the identity "by definition"

In this case, it has two presentations. One as the underlying quasicategory of sSet and one as the underlying quasicategory of sCat, where by underlying quasicategory I mean "nerve of fibrant replacement of hammock localization"

Sure, but those are in principle two different objects. How are you identifying them?

It's a result of, e.g., Hinich, that the Quillen equivalence that N participates in defines an equivalence of quasicategories between these two simplicial sets.

Sure, but then you're using N and it is the identity "by definition"

You're not asking me if an endofunctor is the identity, you're asking me if an equivalence between two different ∞-categories is the identity and I'm not sure that's actually a meaningful question

For a low tech example, can you tell me if an isomorphism between two different groups with three elements is the identity?

So, I see what you're saying. What we have is actually something like Cat_∞-->(uq of sCat)-->Cat_∞

I feel like I'm missing something. What are the two functors? The two components of the Quillen equivalence?

Or rather, we start with a 1-simplex (uq of sCat)-->Cat_∞, which arises from using, say, Dwyer-Kan or Hinich, which is only an endofunctor of Cat_∞ up to equivalence anyway

Since its two ends are (equivalent) vertices in a quasicategory

Actually I think I have to use Hinich's result, not DK

So... I don't know if I'm convincing you that I understand your objections, but you've certainly helped me clarify an issue in my thinking.

late congratulations on becoming a moderator @DenisNardin

::joins Italian Language to show support::

:-)

I see the Greek language proposal didn't make it out of Area 51.

@DenisNardin I think J is making it too complicated

you don't need to bring N into the mix at all

take the BK RelNerve of sCat to get a qCat and take the BK RelNerve of sSet wrt the Joyal equivs

The QEQ of C and N induces an equiv of Qcats by Barwick-Kan

both are models for Cat_infty,

Can you detect if the equivalence has some kind of 'parity' to see if it has a twist (from the ^op) or not

I know that we're implicitly identifying the two

it makes sense if you can somehow attach an "orientation" to both qCats

seems reasonable because they do in fact come from things with natural transformations

@HarryGindi The problem is that you have to specify how you are identifying them. Essentially you are asking whether an equivalence in the ∞-category Cat_∞ is the identity, but this does not make any sense if you don't identify source and target, and the answer depends on the identification

@DenisNardin Yeah I understand that. I was wondering if there was an obvious way to try to do something like you do in homology of manifolds wrt orientations such that ^op reverses orientation

but yeha

It seems like to distinguish them between models, you need to understand the infinity,2 structure

I dunno though

but so... does the thing i said earlier make sense? which I think is basically agreeing with @dhy and @DenisNardin. I mean, the point being that we end up with a 1-simplex between these two vertices in the large quasicategory of quasicategories... and one of them is (by definition) $Cat_\infty$, and one of them isn't.

@JonathanBeardsley I think you can't actually discriminate between the two without sort of "choosing bases", is what Denis is saying, but yeah you do get that

discriminate between id and op.

Is $op: sSet\to sSet$ a right Quillen functor? It obviously preserves fibrations, but it's not obvious to me how to prove that it preserves categorical equivalences.

QEQs are equivs of relcats, and therefore are weak equivs in the huge cat of large relcats, the relnerve of which is a Qcat in which the 1-simplices coming from W are invertible in the relnerve

What Denis was saying was you can't tell if an isomorphism isn't "op" without some way of reflecting that using generators

Ah, well that makes sense I suppose.

yeah ir's just like orientations in singular homology of manifolds

Ultimately, and this is all really silly, I was hoping for a slick way of proving that $N\circ op \simeq op\circ N$, where on one side I'm taking $op$ of simplicial sets, and the other side I'm taking $op$ of simplicial categories.

Basically because I don't think I'm smart enough to actually break open the definition of $N$ itself and check this.

what's your email

I sent an email to Joyal with a generalized version of N

sites.math.washington.edu/~jbeards1

it makes a million times more sense

at least to me, idk

generalizing N to arbitrary wreath products with Delta makes the formula highly regulae

for each hom object

let me know if those notes were helpful in understanding the structure

the composition is exactly the inclusion of the corner tensor

the quoted message has the firsf email

So combinatorially, frakC has a very nice categorical description, even in much more generalitu

Heh, well, considering that I haven't even the most basic intuition for what a wreath product with $\Delta$ looks like, I imagine it might be a bit confusing for me.

in Delta[n](c1,...,cn), Hom(i,i+1)=c_i+1

and it's the free Psh(C)-Cat generated by those generators

Anyway, in the formula I mentioned, assume c_i is the unique obj of the terminal cat

that's isomorphic to Lurie's C

anyway, sorry if ir's more confusing, but figuring it out in more generality helped me understand it way better

Well, yeah I dunno. I'm at the point where I just have this really stupid thing I want to be true and it's obviously true, and it just feels like it'd be a very bad idea to like... spend a lot of time trying to work out some really complicated framework.

I mean, someone on MO should have a proof of this

There is 100% a canned proof

the thing about ^op commuting

Yeah of course. Haha. I guess I'm at the point where I'm willing to embarrass myself in here, but not quite as publicly as MO. But maybe I should just do it.

You want me to ask?

Haha, no no it's okay.

so just ask lol!

Link it and I'll vote ir up

The thing you want to prove looks like it's amenable to a direct proof, fwiw

Yeah, JB, if you compute ^op on C[n], this looks obvious, if I'm not mistaken.