Homotopy Theory

Jonathan Beardsley

01:20

So Lurie defines a tensoring of an arbitrary quasicategory over spaces by saying that, for an object $X\in C$ and a Kan complex $K$, we define $X\otimes K$ to be the colimit of the constant diagram $K\to C$ valued in $X$. Does anyone know where he shows how this interacts with the monoidal structure on $C$, if $C$ has a monoidal structure?

Hm, perhaps we can draw this out under the assumption that the tensor product in $C$ preserves colimits...

Yeah I guess that's pretty straightforward.

Jonathan Beardsley

04:17

If we think of a Kan complex $K$ as an infinity category, under what conditions does $K$ admit finite limits?

I guess it has to be connected, because its needs to have a terminal object.

Harry Gindi

06:05

So, I was wondering

Would anybody find it useful to have a worked out Yoneda lemma/embedding for (∞,∞)-Cats even without the higher Grothendieck construction?

seems like with that, and the native technology in the cellular set/space model, you can develop at least definitions of weighted limit/colimit that are homotopy-invariant as well as also defining lax versions (which shouid also be homotopy-invariant but currently await a proof that the (op)lax functor objects are homotopy-invariant)

I think I can basically construct the embedding and prove it is FF now, up to a few tricky steps that I think I can work with

Harry Gindi

06:27

Just not clear if there is any interest in having that without the fibration theory

It gives you like "model definitions" in terms of representability, but not really any way to compute anything

@JonathanBeardsley When does a groupoid admit binary products?

Jonathan Beardsley

@HarryGindi Yeah I realized I was barking WAY up the wrong tree with that question actually.

Harry Gindi

yep

Jonathan Beardsley

I was trying to think about what "stabilization" of an infinity groupoid should be, but then went and looked at the definition and realized that's not even defined.

Harry Gindi

I think the answer though is "when they are the terminal category" or something like that

Jonathan Beardsley

Yeah I believe it.

Here's a much simpler, and hopefully less vacuous question: if we define the tensoring of spectra over spaces by saying $X\otimes E:=colim(X\to Spectra)$ for the constant functor valued in $E$, why do we get $E\otimes \Sigma^\infty_+ X$?

I think at some point I knew how to think about this.

Harry Gindi

06:41

what is X\to spectra?

Jonathan Beardsley

the constant functor that takes every point of $X$ to $E$.

Sorry, $X$ is a space here.

Harry Gindi

ah

Jonathan Beardsley

(well, Kan complex)

Moreover, I believe that this should hold for the constant functor into any stable quasicategory, where the term $E\otimes \Sigma_+^\infty X$ has meaning because very stable quasicategory is tensored over spectra.

Harry Gindi

I guess consider the case where X_0 is a point?

that colimit is homotopy coinvariants of the action of X on E?

Jonathan Beardsley

Well, I guess if $X$ is connected and pointed then you can think of it as coinvariants of the action of $\Omega X$ on $E$ or something.

What do you mean by $X_0$?

Harry Gindi

06:45

it's a kan complex so I assumed it was actually BG of an infty group

so it only has one vertex

wait hang on, the constant functor X->Spt if X_0 is a point sends all morphisms of X to tbe identity so that colimit is actually E isnt it?

Jonathan Beardsley

Right.

Harry Gindi

that's no good then

Jonathan Beardsley

Er wait, I'm not sure I understand.

You're asking for the Kan complex $X$ to have a unique 0-simplex?

Harry Gindi

yeah

example, the nerve of BG for a nontrivial group G

Jonathan Beardsley

Oh right, but it doesn't even really matter. I mean, yeah, it's the functor that factors through the trivial Kan complex.

Harry Gindi

06:49

Is that really the correct notion of the tensor product?

that looks like it only depends on pi_0(X)

Jonathan Beardsley

Well, that's how, if I understand correctly, Lurie defines the tensoring of a quasicategory over simplicial sets, in Corollary 4.4.4.9 of HTT.

Denis Nardin

@JonathanBeardsley Both functors you described are colimit-preserving functors Space→Spectra sending * to E. We know by some variant of Yoneda that evaluation at * Fun^L(Space,Spectra)→Spectra is an equivalence

Jonathan Beardsley

@DenisNardin yeah, ok. I agree with that. So (and I thought maybe this was where Harry was going with this, and maybe he still is) it really only comes down to checking the value on a point.

Harry Gindi

@JonathanBeardsley I have never seen this before, but yeah that's what it looked like!

I'm not so presumptuous as to try to be socratic with you =]

Jonathan Beardsley

Oh no I didn't think of that way. Just that we were sort of... circling things.

Okay okay, yeah, I get it. All good. I think I even know where the relevant reference for "some variant of Yoneda" is in HA.

Denis Nardin

06:56

Actually it's in HTT, I was just lazy. Let me fish it out

Jonathan Beardsley

Ah, well, I think it's in that exact form, for spaces and spectra, somewhere in the "Tensor products of $\infty$-categories" part of HA?

Denis Nardin

I think he cites HTT for that

But you can replace Spectra with any cocomplete ∞-category

HTT.5.1.5.6 (choosing S=*)

Jonathan Beardsley

Hah, well there it is!

Thanks! :-)

Denis Nardin

(the variant of Yoneda I was mentioning is HTT.5.1.5.3). Glad to be of service!

Jonathan Beardsley

I'm like 99% we've had this exact conversation in the past. Back before I went on a very long 2-categorical detour.

By the way, I asked you (Denis) in here a long time ago about the relationship between the operadic nerve and "op," and I was very stuck on it, but finally (with the help of the extraordinarily bright Liang Ze Wong, as well as this chat room) I was able to get it all written down.

So now I'm back to the original motivation for thinking about that at all, which was Thom spectra...

Harry Gindi

07:06

I'm still fuzzy on the whole operad thing. Does Lurie mostly use them to define homotopy-coherent versions of groups and monoids and stuff?

Denis Nardin

Well, that is a good thing. It's annoying how hard are these things are to write down sometimes

@HarryGindi ∞-Operad== symmetric multicategory

Harry Gindi

mhm

Denis Nardin

(classically operad is used to denote a symmetric multicategory with one object, but I totally agree with Lurie that this is a suboptimal choice)

Jonathan Beardsley

I would say that $\infty$-operad = "category of operators of a symmetric multicategory"

Denis Nardin

Well, that's a model-dependent thing. If you remember the structure morphism to Fin_*, the symmetric multicategory and the category of operators contain pretty much the same amount of data

Harry Gindi

07:10

Is a symmetric monoidal infty category an E-infty monoid in infty-cat though?

Jonathan Beardsley

Yeah I totally agree, there's an equivalence of categories.

Denis Nardin

@HarryGindi Yes

Harry Gindi

and it's also itself an infty operad?

Jonathan Beardsley

I do think that this gets underplayed a lot though. Lurie takes a symmetric multicategory $O$ and constructs a category $O^\otimes$ which is an honest category. And this thing actually has a name!

Denis Nardin

Sort of

There is a functor from symmetric monoidal categories to ∞-operads, but it's not fully faithful if you give the lhs the "obvious" arrows

Morphisms on the rhs are lax symmetric monoidal functors

Harry Gindi

07:12

I saw a talk in Regensburg about this, but I didn't understand it exactly

A talk by Thomas.. something german, his name escapes me

Denis Nardin

Uh... Nikolaus?

Harry Gindi

ywp

he was giving a talk about a paper where he had to solve this problem you mentioned, but it was last November

Denis Nardin

I don't know how familiar you are with symmetric multicategories, but it's a nifty way of speaking of monoidal structures. Really clarifies a lot of things. Unfortunately the treatment in HA is somewhat suboptimal (i.e. a lot of proofs are much harder than they need to be), but that happens

Jonathan Beardsley

Agh wait a sec... what's the left adjoint the forgetful functor out of a slice category?

Denis Nardin

Sliced in what direction?

Jonathan Beardsley

07:20

Well, let's say, in particular, $Top_{/X}$.

Oh actually ugh.

Harry Gindi

product with X

Jonathan Beardsley

Is the forgetful functor even limit preserving there? I think I'm screwing it up.

Denis Nardin

@HarryGindi I don't believe that. What are unit and counit?

@JonathanBeardsley It's not limit-preserving. Products in the slice are fiber products

Harry Gindi

hm, maybe that is the right adjoint?

Denis Nardin

Yeah, that is right

Jonathan Beardsley

07:23

@DenisNardin Right, it's colimit preserving.

Harry Gindi

its right adjoint is the product w X

Jonathan Beardsley

Yeah, ok.

But so, okay, that forgetful functor is also colimit reflecting.

Anyway, I'm just checking that the functor $Top\to Top_{/X}$, for a pointed space $X$, that takes a space $Y$ to the trivial map $\ast:Y\to X$ preserves colimits.

And I think that, more or less, this is the same as colimits in $Top_{/X}$ being created by the forgetful functor.

Harry Gindi

true in any cocomplete category?

seems good

Denis Nardin

@JonathanBeardsley The functor Top→Top_{/X} is just the one sending Y to [Y×X→X]. Then the functor corresponding to "colimit" is just the forgetful functor Top_{/X}→Top, so the composite is just sending Y to Y×X (as it should!)

Tyler Lawson

@JonathanBeardsley you could write it as Top ~= Top/* -> Top/X, the first functor's an equivalence and the second preserves colimits (its right adjoint is taking fibers over *)

your statement about the forgetful functor preserving colimits proves even more, I think

Jonathan Beardsley

07:30

@TylerLawson Oh that's a nice way to think about it.

Harry Gindi

How much data would one need to specify an E_1 monoid structure on an infty-category? Is there any kind of machine that lets you get away with just defining binary products and then freely generating the rest (at the cost of changing the binary products)?

Jonathan Beardsley

Well, if you're freely generating things then you don't need any data, if I'm understanding correctly.

Harry Gindi

well, semi-freely, I dunno

Jonathan Beardsley

Or, I mean another thing to say is that if you've got, say, a monoidal category, then its nerve is an $E_1$-monoidal $\infty$-category.

So that is, in some sense, only defining the binary products, and then just sort of "filling in" the rest.

Harry Gindi

like, you take an infty-cat with a bifunctor and identity for the bifunctor

can you generate a universal A_infty structure

Jonathan Beardsley

07:37

I imagine you could do something by like... writing down a sort of brutally truncated $E_1$-operad, and then Kan extending.

Harry Gindi

I'm thinking about all the problems that Moerdijk and Weiss had with the Dendroidal tensor product

ultimately they had to do brutal combinatorics to show that it was homotopy-associative

Jonathan Beardsley

Like, pick some subcategory of $E_1^\otimes$, which is the "category of operators" of the $E_1$-operad, and (left?) Kan extend along the inclusion.

Harry Gindi

hmm interesting

Jonathan Beardsley

I don't really know though. I mean, what I'm saying is likely ill conceived.

In an ideal world, this subcategory thing would still be an $\infty$-operad, but I'm not sure it is, or even how one would define it.

I mean, what you're saying sounds a BIT like producing the free operad on a binary operation or something.

Harry Gindi

yeah

Jonathan Beardsley

07:41

So I mean, yeah, there are certainly classical notions of "truncated operads," where you have, say, the empty-set, above whatever arity you like.

(e.g. if you want to specify an n-nilpotent operation or something)

Denis Nardin

I don't think Ass has any interesting suboperad

Harry Gindi

Alex Campbell suggested you could do something with specifying a homotopy-associative tensor product without using model cats

Denis Nardin

But yeah, you want to operadically Kan extend along the map Free(2)→Ass, where Free(2) is the free operad with a binary operation

Jonathan Beardsley

@DenisNardin Yeah I was worried about this, if you just chop off the higher arity stuff, does it cease to be an operad?

Denis Nardin

Well, every ∞-operad with one object contains Triv

Or, if you want, in the definition $O_{\langle n\rangle}\cong O_{\langle 1\rangle}^n$, so if the underlying ∞-category is *, all the fibers must be *

Jonathan Beardsley

07:43

Right, good point.

Harry Gindi

mhm

Jonathan Beardsley

But yeah, sounds like Denis idea, of operadically Kan extending from the "free binary operation" operad makes sense.

Harry Gindi

So what is an A_n monoid, does Lurie develop that stuff anywhere?

Jonathan Beardsley

Not that I recall.

I mean, there's a formal machine for taking an operad and producing an $\infty$-operad, for sure.

Denis Nardin

I don't know if Lurie develops it in detail, but an A_n-monoid is just... an A_n-monoid. A_n is a perfectly well-defined operad

Jonathan Beardsley

07:46

Okay, but so there's some way of taking, say, the commutative operad, and forcing all n-fold products to be trivial/contractible/zero right? I seem to recall this.

Denis Nardin

Yeah, sure. But that's not what you want to do

You want to have more n-fold products, not less

For example in the free operad on a binary operation, there are two nontrivial ternary operations, a(bc) and (ab)c

Jonathan Beardsley

Yes I agree. That's not the right way to answer Harry's question.

Harry Gindi

making a noncommutative thing commutative in a universal way seems sort of reasonable, in my mind. You do something like homotopically kill a higher commutator or something

Jonathan Beardsley

I think this somehow gets at the difference between left and right Kan extending.

Harry Gindi

I don't understand what it would mean to make it associative

Denis Nardin

07:48

As long as we're being vague, you kill a higher associator

Jonathan Beardsley

=P

Harry Gindi

heh

Denis Nardin

More concretely, you make the map O→Ass more connected

(this assumes you are given somehow a map to Ass)

Harry Gindi

yep

Oh interesting, just found this

"A version of my second book. Last update: September 2017; rewrote section on the associative operad and added material on A_n algebras."

Jonathan Beardsley

I'm only going to work with "stable, presentable, symmetric monoidal, bicomplete $\infty$-categories" from now on (instead of spectra), to avoid being asked by referees to use the smash product.

Denis Nardin

07:55

Heh. The smash product of spectra is much more similar to the tensor product of modules than the smash product of spaces, so it's a bit unfortunate that ∧ is the standard symbol...

Jonathan Beardsley

I agree!

Harry Gindi

So I was wondering something: Are any of the models for Spec(F_1) the affine spectral scheme of an Einfty ring spectrum?

Denis Nardin

I don't know of any model for F_1 with those characteristics

But then, I don't know of any nice definition of Perf_{F1}

Jonathan Beardsley

Maybe something with $K(\mathbb{S})$ ?

I'm just thinking of the stuff Jack Morava has written. But I don't really have a good understanding of any of it.

Denis Nardin

It also doesn't help that there are at least two different "yoga" of F1 floating around (the base under Spec Z and the residue field at the point at ∞ of Spec Z) and people do not distinguish them. They behave really quite differently

Harry Gindi

08:00

I mean, Thomas Nikolaus mentioned that K theory gives you some kind of universal Lambda-ring thing, and I saw that Jim Borger was working with Lambda-rings to think about F_1

Denis Nardin

Yeah, I don't know. I'm skeptical that K(S) is a good candidate for F_1 but who knows? Not me, for certain

Harry Gindi

So the thing Jonathan just said was kind of the tree I was barking up

Jonathan Beardsley

Hey @DenisNardin, for that thing about $colim(c_X:X\to LMod_R)\simeq R\otimes_R \Sigma^\infty_+ X$, is it enough to notice that you've got two colimit preserving functors out of $Top$ which agree at the point, and that $Top$ is generated under colimits by the point?

Denis Nardin

Wait, no

You need freely generated, unless you already have a natural transformation

Jonathan Beardsley

Haha, that edit was beautiful to watch in realtime.

Denis Nardin

08:03

I really should think before I type

Jonathan Beardsley

Me too!!

But... do you have freely generated in this case? I mean, if I'm understanding what freely means here.

Denis Nardin

Well, basically I mean Fun^L(Space,C)→C is an equivalence for all cocomplete C

Jonathan Beardsley

I.e. $Top\simeq Pr(\ast)$

Right. I mean, I guess I'm just sort of saying the special case of that reference you gave, since $Top$ is freely generated by the point under colimits?

Denis Nardin

If you play around with adjunctions a little bit you see that this corresponds to j_!j=id where j:*→Top is the inclusion of the point

@JonathanBeardsley Yes, exactly

Harry Gindi

@JonathanBeardsley I noticed the answer you gave to your question earlier and saw that you have a paper about it on the Arxiv

Jonathan Beardsley

08:15

@HarryGindi yep, just went up, uhhhhh... Monday?

Harry Gindi

Are you hoping to generalize further?

Jonathan Beardsley

it helped to talk with you about it by the way!

Harry Gindi

like, do you imagine it could work for the base to be a RelCat?

Jonathan Beardsley

I think we forgot to put any kind of acknowledgements in it, but I meant to acknowledge our conversations. I should really acknowledge this room (and probably Denis) in everything I write, haha.

So, for the base to be something like a RelCat, I think you might want to look more at something like Prasma's model category theoretic Grothendieck construction.

Harry Gindi

I'm just wondering because if you could give a version of straightening as the derived functor of the ordinary grothendieck construction

Jonathan Beardsley

08:17

I think what we wrote could probably work (modulo quite a bit of tedium) for something like a base being any category enriched in a Cartesian monoidal category.

e.g. enriched in Set, sSet, nCat.

But, yes, I think you could write it all down for simplicially enriched categories, and get Lurie's Grothendieck construction as the simplicial nerve.

Harry Gindi

I actually doubt that, I say, laying down the gauntlet!

Jonathan Beardsley

Totally plausible.

Harry Gindi

If that were true, it contradicts what I expect for the 2cat case

Jonathan Beardsley

I don't have strong feelings about it. I think Ze, my coauthor, kept saying something about simplicial opfibrations not giving you the "right thing" unless your base was discrete.

Harry Gindi

2-fibrations are way more complicated

yep

Jonathan Beardsley

08:20

In other words, the nerve of a simplicial opfibration over a discrete base is a cocartesian fibration of simplicial sets, but this doesn't work for, say, an enriched base.

Harry Gindi

exactly

Jonathan Beardsley

Ze understands that particular aspect a lot better than I do.

But yeah, that's probably a significant obstruction.

Harry Gindi

That's why I suggested relative cats

Jonathan Beardsley

Ze has work in progress where he's like... seriously generalizing some stuff, like, the entire idea of fibration, but it's all a bit beyond me.

Harry Gindi

I'm thinking of infty cats not as enriched but as homotopical localizations of discrete things

Jonathan Beardsley

08:21

Right. Yeah I guess I don't know!

I'm kind of burnt out on the Grothendieck construction to be honest, haha.

Harry Gindi

I don't know if that's where your interests lie, but extending it to relcats would be really cool

Jonathan Beardsley

I mean, I like USING it, but I'm a bit worn out from like, opening it up and looking at its guts.

Harry Gindi

Yeah. It's rough stuff

Jonathan Beardsley

I don't know where my interests lie anymore. I think my interests lie more or less close to the interests of whomever I'm talking math with.

Harry Gindi

I heard a cool thing though

Jonathan Beardsley

08:23

I used to want to reprove the nilpotence theorem using DAG, but I think that's a bit of a pipe-dream.

Harry Gindi

Nilpotence theorem?

Jonathan Beardsley

Devinatz, Hopkins and Smith's Nilpotence Theorem that says (in one guise) that elements in $\pi_\ast(R)$ are nilpotent if and only if they're trivial in $MU$ homology

Where $MU$ is the complex cobordism spectrum.

It, along with the results that can be proven using it (the so-called Periodicity Theorem and the Thick Subcategory Theorem), is sort of, IMHO, the crowning achievement of chromatic homotopy theory.

Harry Gindi

R is a ring spectrum?

Jonathan Beardsley

Ah, yes, sorry.

But, if I recall correctly, it only needs to be homotopy associative.

I.e. this doesn't have anything to do with operads.

At least not explicitly.

Harry Gindi

Why do you think it's related to DAG? Do you see a connection to deformation theory?

Jonathan Beardsley

08:32

Yes to the latter.

These results basically tell us that the stable homotopy groups of spheres, and in fact the entire category of finite cell complexes, are "stratified" into layers controlled by substacks of the moduli stack of formal group laws.

Harry Gindi

is that a classical stack?

Jonathan Beardsley

Yeah.

It's literally a groupoid in commutative rings.

Harry Gindi

Neat!

Jonathan Beardsley

Yeah it's a pretty long story. But, yeah, I don't know, my thesis, and some work after that, was in a lot of ways about these spectra called $X(n)$ which colimit to $MU$.

and such that $Spec(X(n))$ is a principal $Spec(S[\Omega S^{2n+1}])$-bundle over $Spec(X(n-1)$.

Where you'll notice that $\Omega S^{2n+1}$ is really something like the polynomial algebra on a generator in degree 2n.

Harry Gindi

mhm

Jonathan Beardsley

08:38

And, in the proof of the nilpotence theorem it's shown that each $X(n)$ detects "more" nilpotence.

until you get to $MU$ and you've detected all of it.

Moreover these $X(n)$ spectra appear to have some kind of relationship to "truncated formal groups," which are like formal groups that only have group structure up to the nth infinitesimal "degree."

So there's this whirlwind of interesting stuff going on, and I think being able to say "This is WHY the nilpotence theorem is true" would be pretty amazing, but I may not be approaching it correctly.

So yeah, now I just do whatever's in front of me I guess, math-wise.

Harry Gindi

so how did you end up walking headlong into straightening/unstraightening?

Jonathan Beardsley

Oh lord, haha.

Harry Gindi

Did you need a more explicit model for that stack?

Jonathan Beardsley

If you want to do things like say "Spec" of an E_2-ring spectrum is a principal G-bundle over something else, you better be able to talk about coalgebra and comodules in E_2-rings.

So I wanted to prove that Thom spectra (which are really just quotients of ring spectra by n-fold loop spaces) were in fact comodules for BG, where G is the n-fold loop space you quotiented by.

This is a sort of Koszul duality statement.

Or half of one.

This is analogous to the fact that there is an equivalence between G-modules (in Top) and spaces over BG, where we've just noticed that {spaces over BG} = {BG comodules in Top}, and used the latter formulation because it's more general.

Harry Gindi

mhm

Jonathan Beardsley

08:46

But so anyway, to talk about comodules and coalgebras in quasicategories you kind of can't avoid the Grothendieck construction.

And I have a very specific thing that I want to show is a comodule, which means I need to think about actual models and so forth.

So my "bright idea" was to prove the necessary stuff at the level of simplicial model categories and then take the nerve. But then one needs to check that the nerve preserves coalgebraic structure.

Harry Gindi

so did this new work clarify that computation for you now?

Jonathan Beardsley

Well, this new work basically just lets me say that "coalgebras and comodules in a nice enough simplicial monoidal model category are still coalgebras and comodules in the underlying quasicategory"

It's probably been a horrible waste of time, haha.

Anyway, I very much should go to bed.

Thanks for the chats y'all.

Harry Gindi

goodnight thanks for the chats too!

Saal Hardali

10:17

@JonathanBeardsley When you say that Thom spectra are "quotients of ring spectra by n-fold loop spaces" do you mean this as a generalization of the case where when $X$ is connected the Thom spectrum of a sphere bundle are equivalently the homotopy orbits of $\Omega X$ on $\mathbb{S}$? How does the $n$-fold come about?

user131753

13:42

Let $\mathbf{A},\mathbf{B}$ be two categories and $\mathscr{F}:\mathbf{A}\to\mathbf{B}$ be a non-full embedding such that if $X$ and $Y$ be two $\mathbf{A}$-objects and $f:\mathscr{F}(X)\to \mathscr{F}(Y)$ be a $\mathbf{B}$-isomorphism between them there there exists an $\mathbf{A}$-object $Y^\ast$ and an $\mathbf{A}$-isomorphism $g:X\to Y^\ast$ such that $\mathscr{F}(g)=f$. Does this type of embedding has a name?

Jonathan Beardsley

14:20

@SaalHardali I only mean that if $R$ is an $E_n$-ring then we can talk about actions on it by $E_{n-1}$-spaces via $n-1$-fold loop maps $X\to BGL_1(R)$.

Or, actually sorry, we can actually talk about actions by $E_n$-spaces along maps $X\to GL_1(R)$.

And then of course the quotient is $E_{n-1}$, which manifests as the functor (and hence its colimit) $BX\to BGL_1(R)\to LMod_R$ being (at best) $E_{n-1}$-monoidal.

Shay Ben Moshe

16:47

Hi, does any one know if it is possible to find Morava's preprints on chromatic homotopy?
It seems like a lot of the older papers on chromatic homotopy were referencing them, and it seems interesting to see the original views Morava had on all that.
Some of the preprints I've seen papers referencing: Structure theorems for cobordism comodules, Extensions of cobordism comodules, Cobordism and K-theory, Extraordinary K-theories: Summary
Thanks!

Jonathan Beardsley

18:33

If I had a simplicial monoid $A$, I could do two things to get a monoidal $\infty$-category $Top_{/A}$: I could look at the $\infty$-category $Top=N(sSet^\circ)$ and form the slice over $A$, then notice that this is equivalent to $Fun(A,Top)$, so obtains a monoidal structure as a result of Lurie's Day convolution construction in HA 4.8.1.10.

On the other hand, we could notice that there's a simplicial monoidal model structure on $sSet_{/A}$, and then take the simplicial nerve of $(sSet_{/A})^\circ$, to get a monoidal $\infty$-category. Any idea on how to see that these two are equivalent?

They're both the "Day convolution" monoidal structure, and they MUST be equivalent, but proving it seems a bit non-trivial.

Harry Gindi

@JonathanBeardsley The first thing doesn't sound right

how can you form that slice?

You want to form what Mike Shulman calls the fibrational slice over A, I think

but just Spc/A is just spaces with a map to A

Jonathan Beardsley

18:54

@HarryGindi I'm thinking of something like proposition B.1 here: arxiv.org/pdf/1112.2203.pdf

Although in that case you're getting the symmetric monoidal category with the Cartesian monoidal structure

Harry Gindi

They're doing homotopical localization of the nerve of that actual strict category

that's a presentation of Fib/A

Jonathan Beardsley

How is that different from what I'm doing?

there's a model structure on $sSet/A$ in which the fibrant objects are fibrations over $A$.

Harry Gindi

"$Top=N(sSet^\circ)$ and form the slice over $A$, then notice that this is equivalent to $Fun(A,Top)"

this specific part of the sentence is false without saying that you're doing localization

N(sSet/A)[W^-1] is the same as Fun(A,Top) where W is the class of covariant equivalences

Jonathan Beardsley

I believe that one doesn't need to take [W^-1] if one takes bifibrant objects

As I'm doing, by writing $(-)^\circ$

Harry Gindi

Kan complexes over a Kan complex aren't necessarily fibrations over A though

Jonathan Beardsley

19:00

Every object in $sSet_{/A}$ is cofibrant, the fibrant objects are the fibrations I think.

So the objects of $sSet_{/A}^\circ$ are the fibrations over $A$, I believe.

Perhaps, to be clear, I should write $(sSet_{/A})^\circ$.

Harry Gindi

yep!

Jonathan Beardsley

But I did write that originally I guess.

Harry Gindi

Yeah, in this case, the place where you diverge is by imposing the monoidal product via convolution or by using the fact that you can already take tensor products of functors at the point-set level

Jonathan Beardsley

I mean, there IS a simplicial monoidal model structure on $sSet_{/A}$ when $A$ is a simplicial monoid. And so certainly one gets some monoidal $\infty$-category $Top_{/A}$.

Harry Gindi

I think if you prove that the second construction preserves colimits in each variable and agrees on representables that the Day theorem is a uniqueness statement as well

representables here meaning the representable fibrations A/a for all a in A_0

Jonathan Beardsley

19:08

Yeah possibly... the Day convolution structure is given by an adjunction in HA, so it certainly has a universal property.

Harry Gindi

The classical Day theorem is a uniqueness theorem

Jonathan Beardsley

Do you have a reference for that? It might be nice to look at.

Harry Gindi

I assume the same is true for (infty,1)-categorical version

Yeah, Day's original paper

I think it's on ross street's website

Jonathan Beardsley

Cool. I think I found it in the references at the bottom of the nlab page on Day convolution.

Harry Gindi

web.science.mq.edu.au/~street/Day.pub.html

Brian died young of cancer or something so Street keeps his publication list up on his own page

Jonathan Beardsley

19:10

Also going to check out @SaulGlasman's paper on this

Yeah, Lurie gives a sort of uniqueness statement as well, I think.

Harry Gindi

The full version of Day's theorem proves an equivalence between pro-monoidal structures on A and monoidal biclosed structures on Psh(A)

and every monoidal product on A gives rise to exactly one pro-monoidal structure on A

If the Lurie version has the same thing, then it's basically enough to show that the tensor product on sSet/A preserves homotopy-colimits in each variable, since showing that they are the same on representables is basically by definition.

Jonathan Beardsley

Yeah, I don't see that immediately in what Lurie says, but maybe it's implicit.

Harry Gindi

It's sort of automatic

Jonathan Beardsley

Yeah, so... given an $O$-monoidal $\infty$-category $C$, I wonder if the structure map, the cocartesian fibration $C^\otimes\to O^\otimes$, is a map of $O$-monoidal $\infty$-categories...

Harry Gindi

every Psh X is canonically the hocolimit representables

if the tensor product agrees on representables

and it preserves hocolims

then (hocolim a_i) otimes a = hocolim (a_i otimes a) etc etc

=hocolim (a_i boxtimes a) = hocolim a_i boxtimes a and then rinse and repeat in the second variable and you show that X otimes Y = X boxtimes Y for all X, Y

Jonathan Beardsley

19:30

Hm okay, I think I see what you're saying.

Yeah... okay, so I definitely see why this "must be true." And I think I can even give a sort of rudimentary argument for it...

The rough idea being, as you say, that both monoidal structures extend the monoidal structure on A itself.

Harry Gindi

I think the only thing you have to prove is that the induced tensor product on the model category of simplicial sets sliced over A preserves homotopy-colimits

Jonathan Beardsley

Hm, right.

Harry Gindi

interestingly enough, proving that actually seems to prove it in the case where A isn't even fibrant

Jonathan Beardsley

Oh yeah, I don't know if I said that, but I was definitely assuming $A$ was fibrant.

Harry Gindi

or in the case where we consider either the covariant/contravariant model structure and A is an infty-cat

I think!

(because projectively cofibrant diagrams are cofibrant diagrams in all of those model structures)

maybe that's stupid though

anyway I have to go, seeya!

Jonathan Beardsley

19:40

Yeah, so the preservation of homotopy colimits I think comes directly from the fact that tensoring in sSet itself preserves homotopy colimits. I guess the question is whether or not the forgetful functor from the slice down to sSet, which preserves and creates colimits, also preserves homotopy colimits...

Nevermind, that forgetful map is obviously a left Quillen functor.

Jonathan Beardsley

23:37

This has come up in here before, but does anyone know of a quasicategorical reference for "doctrinal adjunction," i.e. when the left adjoint of a monoidal functor is monoidal, etc?

Maybe sort of Proposition 2.2.1.1?

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