Probably something well known. Has anyone ever written explicitly a theory of Goodwillie calculus for motivic spectra (like studying Goodwillie n-jets)? I mean the generalisation looks like straightforward, but I have never seen anything explicitly written down (people usually stop at degree 1, i.e., spectra), so maybe I'm overseeing something.

@user40276 It depends on what you mean by "Goodwillie calculus". The basic version of Goodwillie calculus works in any presentable ∞-category (and so in particular for motivic spaces), but the more interesting one is not even developed completely for equivariant spectra, that are usually an easier case

I once tried for a while to get a motivic version of Weiss orthogonal calculus, that seems easier. Maybe I should go back to that at some point...

What definition of n-excisive functors were you thinking about?

It would be interesting to see if the slice tower can be gotten this way.

Uh that's a bold conjecture :). Tell me more

The slice tower is a sort of postnikov tower. Googdwillie calculus si supposed to reorganize certain things around connectivity as it is a homotopically meaningful concept whereas dimension isn't.

The only place I know where equivariant calculus is developed is Emanuele Dotto's : arxiv.org/abs/1305.6217 . The missing piece there is the classification of homogeneous functors (he had a nice conjecture with Tomer Schlank in which I believed a lot, but he told me that he has a counterexample)

I don't know if there has been development since then

@DenisNardin What is "the involution $\mathrm{op}\colon \mathrm{Cat}_\infty\to \mathrm{Cat}_\infty$ that carries a quasicategory to its opposite"? (First paragraph of section 1 of "Dualizing cartesian and cocartesian fibrations".)

@CharlesRezk Quite literally, the involution that carries $\mathcal{C}$ to $\mathcal{C}^{op}$ and $F:\mathcal{C}→\mathcal{C}'$ to $F^{op}:\mathcal{C}^{op}→{\mathcal{C}'}^{op}$

What does it do on 2-simplices?

You can define it at the level of the simplicial category $\mathrm{qCat}$, if you want

I tried.

Hmm... so there is an involution $\mathrm{op}:\mathrm{sSet}→\mathrm{sSet}$, we all agree on this, right?

Maybe I'm missing something, but it looks like on simplicially-enriched categories you only get $\mathrm{qCat}\to \mathrm{qCat}'$, where $\mathrm{qCat}'$ has the same objects as $\mathrm{qCat}$, but the function complexes are "$\mathrm{op}$" of the ones for $\mathrm{qCat}$.

Ah I see the problem. Hrmm... probably the quickest thing to do is to define them on complete Segal spaces then

There it literally comes from the involution on $\Delta$

I'm not sure how that helps.

How explicit do you need the definition to be?

I'm basically asking: is there a particular definition you had in mind where you make that statement in the paper?

I know its possible to cook something up that deserves the name.

Well, we know that $\mathrm{Aut}(\mathrm{Cat}_∞)\cong C_2$, we meant any representative of the nontrivial automorphism

Ah.

I don't think we meant any particularly explicit model

Of course, if you want, you can take the functor $\mathrm{Cat}_∞→\mathrm{Cat}_∞$ classifying the op of the universal cocartesian fibration

It's confusing, because I would imagine that one of the reasons you needed to write that paper was that there was no good model for $\mathrm{op}$.

Well, the goal of that paper was, given a fibration classified by a functor $I→\mathrm{Cat}_∞$, to construct a fibration classified by the composition $I→\mathrm{Cat}_∞\xrightarrow{op}\mathrm{Cat}_∞$ without passing via straightening-unstraightening

E.g., one that is literally an involution of the simplicial SET $\mathrm{Cat}_\infty$, and which takes $C\in (\mathrm{Cat}_\infty)_0$ to $C^\mathrm{op}\in (\mathrm{Cat}_\infty)_0$.

You can also construct quite explicitely a fibration over $BC_2$ classified the functor $BC_2→\mathrm{CAT}_∞$ given by the $C_2$-action on $\mathrm{Cat}_∞$, that's quite easy

I don't think the goal of the paper was getting an explicit model of the strict action on the simplicial set $\mathrm{Cat}_∞$. Certainly not my goal

How so?

So, you can construct a simplicial category as follows: objects are quasicategories, mapping spaces are $\mathrm{Map}(\mathcal{C},\mathcal{D})=\iota\mathrm{Fun}(\mathcal{C},\mathcal{D}‌​)\amalg \iota\mathrm{Fun}(\mathcal{C},\mathcal{D}^{op})$

You might need an additional op on the second summand, I don't quite remember. We'll see

I see.

Composition is defined exactly as you'd expect

At some point you'll have to use the isomorphism $\mathrm{Fun}(\mathcal{C},\mathcal{D})^{op}\cong \mathrm{Fun}(\mathcal{C}^{op},\mathcal{D}^{op})$, to define the composition of two morphisms in the "additional component"

This has a map to $BC_2$ sending the first component to the identity and the second component to the generator of $C_2$. You can check that it is a Kan fibration of mapping spaces and satisfies the necessary hypothesis to be a cartesian and cocartesian fibration

OK.

I'm skeptical.

It's been a while since I worked out this construction, so I might be misremembering. What you're skeptical of?

I don't see that composition works out, whether or not you put an extra $\mathrm{op}$ in.

@DenisNardin I'm maybe being a lot more naive. I, for now, only care about the straightforward naive definition of what would be a "motivic n-jet spaces" (i.e., n-excisive functors from finite motivic spaces).

So, let's see how to construct the map $\iota\mathrm{Fun}(\mathcal{C},\mathcal{D}^{op})\times\iota\mathrm{Fun}(\mathcal‌​{D},\mathcal{E}^{op})→\iota\mathrm{Fun}(\mathcal{C},\mathcal{E})$

An $n$-simplex in the source is a pair of maps $\mathcal{C}\times\Delta^n→\mathcal{D}^{op}$ and $\mathcal{D}\times\Delta^n→\mathcal{E}^{op}$

We are going to send it to the composition $\mathcal{C}\times\Delta^n→\mathcal{C}\times \Delta^n\times(\Delta^n)^{op}→\mathcal{D}^{op}\times(\Delta^n)^{op}→\mathcal{E}$

The map $\Delta^n→\Delta^n\times(\Delta^n)^{op}$ should be the one sending $i$ to $(i, n-i)$

@user40276 Then those 1-excisive functors do not recover motivic spectra, I'm afraid. You only get $S^1$-spectra

But that theory has been extensively developed in Higher Algebra, chapter 6

@CharlesRezk Do you see any problem in how I defined the composition?

Is it compatible with simplicial operators?

It.. looks to be, to me. I might be missing something

@DenisNardin Hmm... By finite motivic spaces I mean something constructed by gluing Tate circles and $S^1'$s. So it seems that you recover motivic spectra, but maybe I'm being silly. Sorry if it's completely idiot.

@user40276 I'm sorry, I don't follow you. What's your precise definition of motivic spaces?

With the notation you are using, if $f\colon \Delta^n\to \Delta^m$ is a simplical operator (on vertices sending $x\mapsto f(x)$), then $(f)^{\mathrm{op}}\colon (\Delta^n)^{\mathrm{op}}\to (\Delta^m)^{\mathrm{op}}$ is given by the "same formula": $x\mapsto f(x)$ on vertices.

But $x\mapsto (x,n-x) \mapsto (f(x),f(n-x))$ (second map is $f\times (f)^{\mathrm{op}}$) is not the same as $x\mapsto f(x) \mapsto (f(x), m-f(x))$.

Hrmm... that might be a problem

Luckily I never used that construction

@user40276 To be clear: I don't understand how you are inverting the $\mathbb{G}_m$'s. Of course you get their suspension spectra, but in general they have no reason to be invertible

My intuition is that the failure to have a good and rigid $\mathrm{op}$ involution on $\mathrm{Cat}_\infty$ is ultimately why you needed to write that paper.

@DenisNardin I'm using the usual definition of Nisnevich sheaves on smooth schemes. And requiring a finite motivic space to be something that can be produced by gluing finitely many $S^1$'s and $\mathbb{G}_m$'s.

There is a good involution on $\mathrm{Cat}_1$, the full subcategory of $\mathrm{Cat}_\infty$ spanned by ordinary categories, which you can define ultimately because ordinary groupoids are isomorphic to their opposite.

There is an analogous construction, which converts a Grothendieck fibration over $C$ into a Grothendieck op-fibration over $C$ so that the fibers are replaced with their opposite. (By "Grothendieck (op-)fibration" I mean the classical version of cartesian (co)fibration, I forget what they are actually called.)

@user40276 What functor should represent $\mathbb{S}(-1)$?

So you don't see this problem classically.

@CharlesRezk The only construction of the "fiberwise op" I know in classical category theory is equivalent to ours (and in fact it arrived on the arxiv a little bit after our paper...). You still need to "choose" a (co)cartesian arrow

Oh ok, you're right about that, you don't get a good one classically.

@user40276 Maybe I can tell you where the difficulty comes in. Your story "works" if you work with enriched functors in motivic spaces, but that's in part where the difficulties in setting it up stems from

@CharlesRezk Also, you do have a strict model in complete Segal spaces, so I'm doubtful that this lack of a strict model in q-cats is all that harmful

I'm not saying its harmful. Just reassuring myself that it doesn't exist in qcats.

I'm teaching a course a quasicategories, and revising my notes on the same. I am trying to make sure I understand what is going on.

@DenisNardin Do they really need to be enriched? I mean just take $E (S^m \wedge G_m^n)$ to define the spectrum, where $E$ is 1-excisive from "finite motivic spaces" to motivic spaces

How are you getting the map $\mathbb{G}_m\wedge E_{m,n}→E_{m,n+1}$?

This paper constructs motivic spectra the way you're thinking of, and they use enriched functors precisely for this reason

Also, following the intuition from the equivariant case, we don't expect there to be a Taylor tower anymore, but something more like a Taylor "graph", capturing the complexities of the Picard group of motivic spectra

@DenisNardin I am embarrassed that I had forgotten about Emanuele's work. Thank you for reminding me! I also misspelled Goodwillie. Yeesh.

@DenisNardin Ah!Ok. That was kind of a stupid mistake. I haven't noticed that. Thanks for the paper. I think that's exactly what I was looking for.

The problem of what "higher excisivity" means in a motivic context is, as far as I know, wide open. I mean, we're not really able to finish the job even in the equivariant case, that's usually a lot easier :).

@DenisNardin Just out of curiosity. Why there's no Taylor tower? There's no way to construct a Weiss topology or something like that?

@user40276 Equivariantly what happens is that you get a notion of "I-excisive" for each finite G-set I. There are various implications of the form "I-excisive ⇒ J-excisive" for various more or less complicated relationships between I and J, and the resulting poset is a lot richer than a tower

When G is the trivial group this reduces itself to the classical Taylor tower (because there aren't that many types of finite G-sets :)) but in general it's a mess

You get several subtowers of this poset, of which the most important is probably the one given by "nG-excisive", which is somehow "cofinal" in the poset

That's more complicated than what I thought. But what do you think that would be notion of "excisiveness" in the case motivic spaces (what would be the analogous of a G-set in this case)?

I have no clue :)

As I said, it is wide open

Behind the equivariant case stands the realization that finite G-sets are the "building blocks" of G-equivariant homotopy theory. As far as I know, there's no corresponding notion for motivic spectra

E.g. no statement corresponding to the tom Dieck splitting

The reason I wanted to try to attack Weiss calculus is that in the motivic setting we have at least a reasonable notion of vector bundle you can start from

But Goodwillie calculus? Who knows

This may be silly, but isn't C_2 equivariant stuff like motives over R? So wouldn't something like a G-set for G some Galois group be a reasonable guess?

Well, yes and no. C_2-equivariant stuff is very closely related to motives over a real closed field, yes, but it is a lot easier because there you only have the "obvious" generators (C_2/e and C_2/C_2), while R and R[i] are not nearly enough to generate SH(R)

So you get a map SH(R)→Sp^{C_2} that "discards all the other stuff". And that's helpful! But it doesn't help us seeing the deeper structure in SH(R) that we'd need to decide what "higher excisivity" means

Said it differently: the equivariant cohomology of X(C) is still quite far from the motivic cohomology of X

I'll say as much: the Hopkins-Morel theorem suggests that quadratic forms might be involved. But that's all I'm willing to say for now

@DenisNardin Ok. Cool. Thanks for the clarifications. I would still somehow like if a Galois-like think was related to it (like a motivic Galois group), but maybe I'm just being naive and overly optimistic.

No one really knows, so feel free to make conjectures (I have my own :)). Just expect them to be probably wrong

I have a question for the room: let M be the ring of infinite matrices with integer coefficients with finitely many nonzero terms in each row and in each column. Then we can construct the ring spectrum $R=End(\mathbb{S}^{\oplus \mathbb{N}})\times_{\pi_0End(\mathbb{S}^{⊕\mathbb{N}})}M$. The ring $M=\pi_0R$ has an involution sending a matrix to its transpose. Can I lift this involution to $R$?

@DenisNardin @CharlesRezk for what it's worth Lurie, in the proof of SAG.14.4.1.1, encounters this issue and decides to model Cat_infty by topologically enriched categories, where the op-ing isn't problematic.

@DenisNardin I was working on an explicit model for something, and I found completely by accident another proof of parametrized straightening when the parameter ∞-category is an ordinary category lol

@DenisNardin The "classical" construction to which you refer is due to Bénabou in the 1970's.

unmarked parametrized straightening

@AlexanderCampbell Good to know, thanks! Then the person putting that paper on the arxiv really didn't check the literature :)

I think I misunderstood the result in the paper, but when I unraveled what \underline{Top}_T was, I ended up with exactly the thing I was working with

Did you guys ever work anything out about what happens with parametrized straightening when the parametrized ∞-category X is a 'segal category object' (and so represents a sPsh(T)-enriched ∞-category?

Not really. Do you expect something special to happen?

well the inclusion of segal category objects has a right adjoint, so you do get something interesting

so there should be an 'enrichification' of \underline{Top}_T

It's probably just going to be Top_T with the canonical enrichment

That's what I'm hoping for

when I was trying to work it out in explicit models though, I couldn't figure it out

Dunno, all the examples we care about are pretty far from enriched categories, so we never really thought about that adjunction

well, it looks a lot to me like it's the key to higher straightening

when T is, say, Theta

Good to hear :). I meant "we" as in "the people working in that project". But if you have other applications, even better!

I should probably mention that the construction of \ul{C}_T was already in Gepner-Haugseng-Nikolaus; even if we didn't realize it until after we wrote that section

We arrived at essentially the same property from quite a different perspective

In explicit models, this ends up being Fun(T/t, Top_T)

Yeah, that was our motivation

Since (O_G)_{/G/H}=O_H

Cool!

I came upon this construction by a completely different route, so it's very canonical

Yeah, it seems to pop up in several different places.

I think you missed an "op" though :)

probably

Anyway, I'll try to see if I can use what you guys have already done instead of doing all this explicit model stuff

Anyway I'm leaving the internet for today. If anyone has a suggestion for my question above about the ring with involution, please ping me :)

gn

Thanks, you too