It seems 100% sufficient to compute that op(C[n]) = C([n]^op) and show that they define isomorphic cosimplicial objects, and you're done

It's not a hard combinatorial proof. You can do it!

@JonathanBeardsley yeah I figured it out while driving. I can fill in details if you want.

@HarryGindi So the $n$-simplices of $N(D^{op})$ are $Hom_{sCat}(\mathfrak{C}[n],D^{op})$. Are you using the fact that the whole simplicial set is the mapping simplicial object between cosimplicial simplicial categories, and taking the constant cosimplicial simplicial category in the right coordinate?

[n]->N(C)^op is the same thing as a map [n]^op -> N(C), same thing as C([n]^op)->C

using the isomorphism of cosimplicial objects

that's the same thing as a map C([n])^op->C, same as C[n]->C^op, same as a map [n]-> N(C^op)

But I mean, so you've shown that the two things have the same n-simplices, but have you shown that this is all coherent with the face and degeneracy maps?

the iso of cosimplicial objects means naturality

so it's on-the-nose natural

Okay, yeah, I mean, I'm going to have think about this I guess, because [n]^{op}=[n], so somehow... I'm missing something.

right, but [n]^op is not equal to n as a cosimplicial object

err [n]

I guess I'm just very confused about how you're saying anything about the entire simplicial set if you're not producing it, in one go, as the mapping space between two cosimplicial objects. But whatever, I dunno. I'm having a very bad day with this junk lol.

It just seems like this argument is all about the sets of n-simplices. Which is the trivial part.

For any simplicial set, S, Hom([n]^op,S) viewed as a simplicial set is isomorphic to Hom([n],S^op), right?

you don't need to form function complexes

Is [n] here supposed to be $\Delta^{n}$?

yeah

Okay. Sure. it's self-adjoint.

so that's all I've applied the whole way through on that part

the only part where you have to check is that the thing I said about C([n]^op) is naturally isomorphic to C[n]^op

everything else is formal

And checking that is a matter of just understanding what C does on simplices, and what the op in a sCat is doing, and that's a calculation to construct the iso if you want

the iso is going to be the identity literally everywhere in sight

just have to show that the faces and degeneracies work out the same

Yeah, I mean, I'm not actually that worried about that part.

then what's the thing that worries you?

But maybe I'm just failing to understand how the simplicial nerve even uniquely defines a simplicial set, because it looks like it's only defining the n-simplices.

And I think this is just a basic, like, simplicial set fact that I'm being stupid about.

because C is the left Kan extension of a cosimplicial object

Perhaps the fact that it's secretly using the cosimplicial object $\Delta\to sSet$ given by $[n]\mapsto \Delta^{n}$.

Hom(F, X) if F is a functor Delta-> C, defines a functor Delta^op->Set, right?

You don't even need to use the fact that C is left Kan extended. All of the action happens only in the simplex category

this is confusing, I used C too many times

lol no i mean, i'm following it by context actually

so for the record i really do think that the simplicial set you're getting can be written as coming from the simplicial enrichment on cosimplicial objects, where you take a constant cosimplicial simplicial category on one side

sure, that's the definition of a 'nerve'

that's, i think, another way of saying "Hom(F, X) if F is a functor Delta-> C, defines a functor Delta^op->Set, right?"

yep

in which case, sure, defining an iso of cosimplicial objects between $\mathfrak{C}[\bullet]$ and $\mathfrak{C}[\bullet]^{op}$ gives you the result.

yep

and that iso happens to be the identity everywhere in sight

yeah

so they're actually equal

scary term but yeah

hah. well, i've been teaching introductory logic this quarter, and had to use "equal" a lot

i'm inured to it at the moment

Adeel Khan likes to use equal for isomorphic and isomorphic for categorically equivalent

in his classes

which I found fun and amusing

hah

anyway, goodnight!

Later!

cya

@user2236 can you stop? or leave?

is this a private room?

Well I guess I'll have to go look up what the rules are for banning trolls...

ask @DenisNardin

and don't call me a "troll"

I mean... everyone in here is pretty chill, and there are a bunch of anonymous "users" showing up lately saying generally creepy things...

And not talking at all about homotopy theory...

So I dunno, seems like it's probably a good idea to kick you.

I've had one person make a personal comment about my family, another one keeps pointedly talking about the weather in the Pacific Northwest when I'm around (which is where I live), and now you.

I have never interacted with you.

lol

Also people for some reason bothering Denis, which is really irritating.

chat.stackexchange.com/rooms/81673/…

@user1732 haha thanks! we had no idea if that'd actually find its way to the internet...

@JonathanBeardsley any quillen equivalence determines an adjoint equivalence of quasicategories. (and any equivalence can be upgraded to an adjoint (equivalence)). i'm not sure what you mean by "Quillen equivalences induce equivalences after (co)fibrant replacement" though, i feel like that statement is mixing category-levels

@JonathanBeardsley if nothing else, this follows from the fact that \frakC is a left quillen equivalence so creates weak equivalences among cofibrant objects (and all objects are cofibrant, in particular quasicategories are). i guess also you need to know the fact (proved in HTT) that the three definitions of "hom-sset" introduced in chapter 1 are all weakly equivalent to the one you get via \frakC

@IlaRossi i would imagine that this is in goerss--jardine? ultimately, this is just coming from the fact that homotopy groups are defined to be maps in (from spheres), and you only are "supposed" to map into things that are fibrant -- which in this case means kan complexes

@JonathanBeardsley earlier than this, i'm pretty sure it was proved by dwyer--kan in one of their papers around '80 and '81

@HarryGindi i don't know if i would say that "most" relative categories are fibrant. it was proved by lennart meier that model categories are Barwick--Kan fibrant (iirc without any further adjectives necessary)

@JonathanBeardsley what?! i really liked that picture! i wonder why they removed it

@HarryGindi i don't know about general PDEs, but certainly D-modules are relevant in the homotopical world

@AaronMazel-Gee you're very welcome my friend, and thank you for representing us at the ICM.

@JonathanBeardsley yeah, that's an equivalence so its left adjoint is itself, and certainly it is a left quillen functor

(in case it wasn't clear, i'm catching up from like 2 weeks ago)

err, it's an involution so its left adjoint is itself

@AaronMazel-Gee Did I say most? That's definitely not true since W should be Thomason-fibrant. My mistake if I said that

I still find the whole Thomason model structure highly mysterious

like, hocolims are the Grothendieck construction.

What"s still unclear to me is if that determines the Grothendieck construction up to equivalence of fibred categories, or only up to Thomason equivalence, which seems way weaker.

I wonder if Meier's theorem applies to DHKS homotopical categories admitting a 3-arrow calculus

@HarryGindi oh interesting, thomason-fibrancy of W is a necessary condition for BK-fibrancy of (R,W)?

i also find the thomason model structure mysterious. i set up a less mysterious (and pretty straightforward) analog for $\infty$-categories in the fappendix here: arxiv.org/pdf/1510.03525.pdf

as for the grothendieck construction computing hocolims, i think the more fundamental thing is that the grothendieck construction itself is a lax colimit. combining this with the fact that ($\infty$-)groupoid completion is a left adjoint, you immediately get that $|Gr(F)|$ is the colimit of $B \xrightarrow{F} Cat \xrightarrow{|-|} Spaces$

@AaronMazel-Gee Yep. That is, I assume, the only condition. That's the condition that lets Sd^2(W) to be a Kan complex.

@AaronMazel-Gee hm right so then I think we can get that op and N commute up to homotopy by using the uniqueness statement from Barwick-Schommer-Pries

@JonathanBeardsley If you want to go that route, I guess you still have to prove that ^op_s and ^op_Delta both lie in the unique nonidentity component of Aut(N(Qcat)) and Aut(N(sCat)) whatever nerve you mean in this particular case (the B-K relative nerve has the advantage here bc sCat is not a simplicial model cat)

I think the direct proof has a lot of advantages here, since it gives a point-set on-the-nose isomorphism

Sure but TBH I still don't see how all those isomorphisms commute with the face and degeneracy maps

But I also am ok with admitting that that is probably just my general lack of mental acuity

Yeah, I left a comment on my answer to that effect. The "immediacy" of that calculation really depends on the construction

No lol Jonathan

Luckily I have a co-author who's basically a category theory genius. So I'll ask him next time I see him lol

If you only read Lurie's description of that functor

I cant see how you could make heads or tails of it

Joyal has a paper where he works out allll the combinatorics of the coherent realization

So do Riehl and Verirty (they call it a simplicial computad)

Lurie's description is ostensibly lifted out of Joyal's paper without explanation of what the heck it actually is

google.com/url?sa=t&source=web&rct=j&url=http://…

I hate google's stupid link hiding thing

that's a pdf link

"Quasicategories vs Simplicial Categories" by Joyal

Yeah my co-author has been using Riehl and Verity's stuff to better understand C[n]

I know that Dugger and someone wrote a paper using necklaces to understand that thing. I wonder if I wrote up my notes on the enriched coherent nerve (that I emailed you) if that's publishable

I kind of need to get something published if I want to do a Ph.D in Australia, since they require that or an extra year of the master.

I dunno

Hm, where are you now?

Regensburg

Well, I am in NJ now visitng my family for holidays

Oh right, I think I knew that you were in Regensburg. Cool, lots of great math going on there.

Yeah, definitely, but I'd like to stay and work with Cisinski on the Ph.D if possible, but I'm trying to keep options open

not put all my eggs in one basket, as it were

I mean, I'm open to coming back to the US too, but I don't have any ideas for advisors here who are interested in higher straightening/higher Yoneda, which I am convinced is the big open problem for infinity, n-cats

Gaitsgory and Rozenblyum, I guess, but I think they're more interested in applications of those ideas vs actually getting a hold of them in full generality

@JonathanBeardsley Don't sweat it. As it was mentioned I have now mod superpowers, so s/he can do very little to upset me. Since you're the room owner, let me know if I can be of any assistance here with the moderation (moderators on SE have network-wide chat moderating powers, but this is not my turf, so to speak).

oh yeah btw, Denis, congratulations

That said, it is better if we leave further discussion of the moderation aspect for a private chatroom.

@HarryGindi Thanks, but it's really not much worth mentioning. Italian.SE is a very small site, and they needed the help so I volunteered.

but you're the first regular in this room to have mod superpowers

@DenisNardin Oh so like, you're some kind of special MO person now?

i think @EricPeterson is the first to publish?

Did you win some kind of election?

@JonathanBeardsley No, this works only in chat sorry. I am a moderator on Italian.SE (see italian.meta.stackexchange.com/questions/1341/…)

That is a beta site, so there's not a formal election, just some sort of selection by acclamation

he was "appointed" :P

Regardless I have chat superpowers, so I can, for example, suspend people from chat, detect when multiple accounts belong to the same person etc.

Oh cool.

I'm not allowed to mention most things we can see, but again if you need help I'm here

No worries, do your thing!

@DenisNardin have you guys discussed merging Italian.SE with Latin.SE?

cryptic

indeed

Installing TeXLive here on holiday

It always amazes me how long this takes

It always amazes me how good it looks :-)

(compared to pencil & paper)

I guess I'll post this here, since Harry and I were already talking about it:

@MartinSleziak you could have easily won the mod election :-)

@user1732 I am not sure about that. But that would definitely decrease the quality of the moderator team.

Since you mention that there is no longer an intersection between MO and MSE moderators.

right

@MartinSleziak I thought you were going to open up a ICM2018 news room like the World Cup room?

lol

It would probably have close to zero activity. To be honest, I did not follow it very closely - the topics discussed there are far out of my league.

the theft of the fields medal was a bit unusual

not to mention the victim being a refugee

all and all an inspiring story :-)

i really like how he laughed it off

It's enough to show everything works for generating cofaces and codegeneracies

the codegeneracies are free, the 0 and nth cofaces are free

all of those can be done treating frak{C} as a black box

the only slightly complicated thing is keeping track of the inner generated cofaces, but if you use my description of frak{C} or the one Joyal uses in the quasicategories vs simplicial categories paper, the combinatorics are completely explicit for codimension 1 face inclusions

the maps on vertices are obvious, and the maps on homs are just appropriate inclusions of cubes on the {0} face of the cube wrt the axis corresponding to the omitted inner vertex

In general, each Δ[1] factor in Hom(i,j) corresponds exactly to a vertex k with i<k<j, so omitting k gives inclusion onto the 'bottom' face wrt that axis, i.e. Δ[1]^{k-i-1} x {0} x Δ[j-k-1] (I'd call this the top, but I seem to draw my cubical diagrams in the reversed orientation).

sorry, not Δ[j-k-1], rather Δ[1]^{j-k-1}

@PraphullaKoushik Hi, I have noticed that you most of your questions are without a top-level (arXiv-like) tags. For example, the most recent one: To check if a stack is coming from a manifold has only the tag stacks.

It is recommended to use top-level tags for several reasons - see meta: Frequently asked questions about tagging on MathOverflow and Why are MO tags formatted as they are?.

Using such tags is useful for searching and organization of the site. quid's answer to Frequently asked questions about tagging on MathOverflow explains that there are also advantages for the OP.

> Thus, using appropriate tags one can increase ones chances that users competent to answer the question, or just interested in it, will notice the question in the first place. Conversely, using only very specialized tags (which likely almost nobody specifically favorited, subscribed to, etc) or worse just newly created tags, one might miss a chance to give visibility to ones question.

I am not sure to which extent this effect is noticeable on smaller sites (such as MathOverflow) but probably it's good to follow the recommendations given in the FAQ. (And MO is likely to grow a bit more in the future, so then it can become more important.) And also some smaller tags have enough followers.

You are asking posts far away from areas I am familiar with, so I am not really sure which top-level tags would be a good fit for your questions - otherwise I would edit/retag the posts myself. (Other than possibility to ping you somewhere in chat, the reason why I posted this in this room is that users of this room are likely more familiar with the topics you're interested in and probably they would be able to suggest suitable tags.)

I just wanted to mention this, in case it helps you when asking question here. (Although it seems that you're doing fine.)

Hahaha damn. I was just thinking how it might be easier to deal with the coherent nerve everywhere in sight if we just used categories enriched in cubical sets

Lo and behold, there's a paper of Kapulkin and Voevodsky from this year

math.uwo.ca/faculty/kapulkin/papers/…

@MartinSleziak even I was not sure what other tags are appropriate to add.. I will see other questions similar to this, see what tags they have added and will add if I get to see any relevant tags.. thanks for your suggestion.. it is very reasonable,.

@PraphullaKoushik Possibly (dg.differential-geometry)? You seem to be very interested in the differential side of things at least in my opinion

I guess it would be too general tag for that... @DenisNardin if I can not think of anything else then I would add this tag..

Let me put this link again, in case you missed it: meta.mathoverflow.net/q/1457

You don't need to put only one tag, you can put up to five. In general it is recommended to put a very general tag (usually an "arxiv" tag) to indicate broadly which sector of math your question is in, and then more specific tags

@user2646 there are people who regularly visit this room with papers written before i was born

@EricPeterson :pics or it didn't happen.

@SeanTilson I saw you are running European Talbot?

How do I apply next year and also leverage the fact that we've spoken before? 😉

Lol.

I saw it on a poster in the dept

There are multiple organizers and this was my last year. I suggest you wait until there is an announcement for the next one which probably won't be until January.

wow RIP

I would also point out that it isn't necessarily to your advantage to have spoken with me.

Why RIP?

RIP my potentially unfair advantage

Oh, how thoughtful of you.

I thought you were implying that the current organizers might not be up to the task, which is certainly not the case.

No, I met two of them in Cambridge and they're lovely!

One of them was named Jack and I forget the other one's name but she was also very nice!

Kinda weird how many of the European Talbot organizers are in homotopy theory, though!

Lol, why is that weird?

Alice Hedenlund is awesome.

Like... how do you think Talbot's work?

@SeanTilson I dunno, I was under the impression that the US Talbots are usually not focused on homotopy theory!

Ok... and?

And nothing =)

I would say that the topics of the US Talbot, as with the European Talbot, are heavily influenced by the organizers. If you look at who the organizers were/are for the US Talbot I think you will find many homotopy theorists among them.

The US Talbot is definitely focused on homotopy theory

Look at the past years' topics:

2018 - Riehl, Verity's work

2017 - Obstruction theory for ring spectra

2016 - Kervaire invariant one

etc.

Yeah the Talbot has traditionally been (at least) algebraic topology, usually with a pretty serious focus on homotopy theory.

The ones before the ones Denis mentioned are chromatic homotopy theory and Goodwillie calculus

@EricPeterson Haha, yeah let's see... Charles, Tyler, Chris S-P... probably more people I'm forgetting.

Oh wow! That's surprising.

Hey, does there exist a fibrant replacement functor for the Joyal model structure that preserves cartesian products?

There are two for the Quillen model structure, namely Sing|°| and Ex^\infty, but those don't seem to be good.

@HarryGindi what is the definition of a fibrant replacement functor? Some monad of a quillen adjunction?

@AliCaglayan An endofuctor F together with a natural weak equiv Id->F and such that F(X) is fibrant for all X

When Lurie constructs the Yoneda embedding, he needs a product-preserving fib replacement to land in Kan

He uses Sing|°|, but he could just havr easily used Ex^infty

You have a pair of adjoints where one is the nerve to sset quillen right

Perhaps you can then use the quillen one and come back

That would be my first try

So sset joy to sset quil then sing |o| and then back to joy

no

Definitely not. That would make it a groupoid

The reason why Sing|°| works is that the geometric realization and Sing commute with products and all spaces (cgwh, whatever your category of spaces is) are Serre fibrant

I am not aware of any existing model for Cat_infty that is even right-proper, let alone one in which all objects are fibrant