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added to n-excisive functor a section
Is it worth mentioning that topos-ness and cohesiveness are retain by the jet construction at jet (infinity,1)-category?
Sure, here.
(I am a bit time pressured with plenty of other tasks. If anyone feels like expanding further, I’d very much appreciate it.)
I have fixed the paragraphs on spectrum objects as reduced excisive functors at n-excisive functor – Examples – Goodwillie n-jets and at spectrum object – Definition – Via excisive functors which both had minor but crucial (and complementary) omissions in the assumption clauses.
In Goodwillie’s Calculus II: Taylor series, theorem 1.13 concerns the convergence of the Goodwillie Taylor series. There must be a versin of this also in section 6 of Lurie’s Higher Algebra, but where?
He seems to refer back to Goodwillie
We refer the reader to [59] for a treatment of these types of convergence questions. (p. 757)
I guess the renumbering within Higher Algebra is going to require quite a few changes on the nLab.
Thanks. I haven’t checked the renumbering in detail yet. Will have to do so at some point for the sake of the page Higher Algebra. Hm.
I used to have the impression (and we had had corresponding discussion here) that for $\mathbf{H}$ an $\infty$-topos then the sequence of $\infty$-toposes of $n$-excisive functors
$\cdots \stackrel{\longleftarrow}{\hookrightarrow} Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \stackrel{\longleftarrow}{\hookrightarrow} Exc^{n+1}(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \stackrel{\longleftarrow}{\hookrightarrow} \cdots$which starts out with the tangent $\infty$-topos
$\mathbf{H} \hookrightarrow T \mathbf{H} \hookrightarrow \cdots$approximates the $\mathbf{H}$-Sierpinski (∞,1)-topos (parameterized objects in $\mathbf{H}$)
$\cdots \hookrightarrow \mathbf{H}^{\Delta^1} \,.$But now I realize that I don’t know why this is true, if it is true. Probably it’s more like parameterized objects with a section. (?) What is $\underset{\longrightarrow}{\lim} Ext^\bullet(\infty Grpd_{fin}^{\ast /}, \mathbf{H})$?
and we had had corresponding discussion here
There was the discussion here that includes what was added to twisted cohomology about jet toposes.
I see at the end I’m wondering why the jet toposes are differentially cohesive and yet the Sierpinski topos isn’t when the former are supposed to approximate the latter. Is that a sign that they don’t?
There was a little more discussion here.
One thought I tried is to consider the left Kan extension along the inclusion
$\{ \ast \to S^0 \} \hookrightarrow (\infty Grpd_{fin}^{\ast/})$hence the induced coreflective inclusion
$\mathbf{H}^{\Delta^1} \stackrel{\hookrightarrow}{\longleftarrow} [\infty Grpd_{fin}^{\ast/}, \mathbf{H}] = \mathbf{H}[X_\ast]$of the Sierpinski topos into the pointed object classifying topos. That inclusion takes a parameterized object $(E \to X)$ to the presheaf which sends a pointed homotopy type $\ast \to S$ to the pushout-product of the two morphisms.
Unfortunately, while these functors
$(\ast \to S) \mapsto (E \times S) \underset{E}{\coprod} X$preserve homotopy pushouts, they don’t preserve homotopy pullbacks, and so I am not reduced to the higher cubical Blakers-Massey theorem for showing that they are analytic. Need to try something else…
I probably ought to be looking at the right Kan extension, rather, the direct image of the induced geometric embedding. That takes …hm… an object $(E \to X)$ in the Sierpinski topos to the functor
$S \mapsto \left\{ \array{ \ast &\longrightarrow& E \\ \downarrow && \downarrow \\ \underset{i.e. [S,\mathbf{2}]}{[S,S^0]} &\longrightarrow& X } \right\}$Hm.
Sorry: in the above, if I speak of geometric embedding above, then I need to use not the Sierpinski topos of bundles, but the topos of bundles with sections.
I had failed to respond to this here
why the jet toposes are differentially cohesive and yet the Sierpinski topos isn’t when the former are supposed to approximate the latter
Meanwhile David and myself have talked about this by email, but I’ll say it here again, for the record.
What the jet toposes $T^{(n)}\mathbf{H}$ approximate, taken at face value, is $\mathbf{H}[X_\ast]$. That is infinitesimally cohesive over $\mathbf{H}$, because it is $\infty$-presheaves on the site $\infty Grpd_{fin}^{\ast/}$ which has a 0-object.
Now the Sierpinski topos $\mathbf{H}^{\Delta^1}$ canonically maps to $\mathbf{H}[X_\ast]$, but not fully faithfully so. Hence the Goodwillie tower construction does not in any evident way just restrict to the Sierpinski topos.
But the “sectioned” Sierpinski topos fixes that, the $\infty$-presheaves on the generic section diagram $sec \coloneqq\{id \colon \ast \to x \to \ast\}$,which are bundles $E \to X$ in $\mathbf{H}$ that in addition are equipped with a section. This $\mathbf{H}^{sec}$ does have a fully faithful embedding into $\mathbf{H}[X_\ast]$. And since $sec$ does have a 0-object, $\mathbf{H}^{sec}$ is infinitesimally cohesive over $\mathbf{H}$.
So this suggests that $\mathbf{H}[X_\ast]$ is the important thing, and that $\mathbf{H}^{sec}$ is an approximation? And why do we care about the latter, because of it being a form of twisted cohomology?
What is the cohomology of $\mathbf{H}[X_\ast]$?
Also from our discussions there’s how to think of $\mathbf{H}[X_\ast]$ as about adding a “nothing-anti-modal type”. Should we be wondering how the Goodwillie tower fits with the cohesive modalities?
So this suggests that $\mathbf{H}[X_\ast]$ is the important thing, and that $\mathbf{H}^{sec}$ is an approximation?
Yes.
And why do we care about the latter, because of it being a form of twisted cohomology?
Yes, because the intrinsic cohomology of $\mathbf{H}^{\Delta^1}$ is the twisted cohomology of $\mathbf{H}$.
What is the cohomology of $\mathbf{H}[X_\ast]$?
I was wondering, too. Of course it is what it is, and maybe it just doesn’t have an established name yet.
Should we be wondering how the Goodwillie tower fits with the cohesive modalities?
It just runs in parallel, as for Smooth global equivariant cohesion:
So as a cohesive $\infty$-topos over $\infty Grpd$ then $\mathbf{H}[X_\ast]$ has cohesive modalities that factor as
$\sharp \simeq \sharp_{smooth} \circ \sharp_{parameterized} \simeq \sharp_{parameterized} \circ \sharp_{smooth} \,.$Now $\flat_{parameterized} = P_0$ (the 0-th Goodwillie projection), and so now there is this tower of $(\flat_{parameterized})_n \coloneqq P_n$. Not sure yet how to best think of this as part of the big picture.
A vague thought late in the day, but is there something relating your propositions as projections thought with the appearance of sections in $\mathbf{H}^{sec}$. I mean if jet toposes are models for some part of dependent linear type theory
Where tangent (∞,1)-toposes are the archetypicals models for linear types depending on non-linear types (this we consider below) the archetypical model for linear types depending on other linear types might be higher jet (∞,1)-topos. This remains to be thought about,
and if $\mathbf{H}^{sec}$ approximates what jet toposes converge on, and if quantum logic embeds via linear logic into dependent linear type theory, then we shouldn’t be surprised to find split monomorphisms or sections about.
Hmm, is there a notion of parameterised split monomorphisms which might warrant a row in the twisted generalized cohomology in linear homotopy type theory – table?
That’s a good point: via the smash product of pointed objects, $\mathbf{H}^{sec}$ is already a genuine (i.e. non-Cartesian) model for dependent linear homotopy type theory, where $\mathbf{H}^{\Delta^1}$ is not.
Right now I am not sure if there is anything of substance to be said about split monos in this context, but I should think about it, good point.
In fact that’s example 3.9 in Quantization via Linear homotopy types (schreiber), being the evident $\infty$-version of 12.3 and 13.7 in Mike’s Framed bicategories and monoidal fibrations and 2.33 in his Enriched indexed categories.
Not a huge point, but to remind people that parameterized homotopy theory of spaces over a base with a section calls such spaces ex-spaces.
In my HHA paper I make a small generalisation to spaces over a base equipped with a homotopy coherent family of local sections, which seems more natural from a homotopy/higher category POV, but not sure sure how it fits in with the above picture.
What of split monos in general within dependent linear type theory. Shouldn’t there be something like collapse of the wave function in Lagrangian quantum field theory?
David R., re #19: that’s of course true, I should have thought of speaking of ex-objects above.
I wonder what to make of some of the issues in the discussion above, now we’ve seen where things have headed, e.g., in Urs’s recent talk Effective Quantum Certification via Linear Homotopy Types, slide 62:
The key hint for how to progress came from developments in higher topos theory:
Joyal (2008), Hoyois (2016) show unification of classical and linear homotopy-types: remarkably: ∞-categories of bundles of linear homotopy types are again ∞-toposes!
So that’s alluding to fact that the tangent ∞-category of an ∞-topos is an ∞-topos. But above we’re talking about a broader range of constructions for an ∞-topos, $\mathbf{H}$, the further ∞-toposes: $T^{(n)}\mathbf{H}$, jets; $\mathbf{H}^{\Delta^1}$, Sierpinski; $\mathbf{H}^{sec}$, presheaves on the generic section diagram; $\mathbf{H}[X_\ast] = [\infty Grpd_{fin}^{\ast/}, \mathbf{H}]$.
Presumably there are useful things to say about these from the perspective of Linear HoTT.
(By the way, there’s a typo in the slides: “the potential is inderminate”, slide 204).
Okay, I have finally fixed that remaining slide! :-)
Regarding your question: Is there is good smash product on $n$-excisive functors for $n \geq 2$?
Incidentally, I am looking for good model category of unbased 1-excisive functors (i.e. modelling parameterized spectra) which would be both type theoretic and doubly monoidal in a suitable sense. (The ones currently mentioned at model structure for parameterized spectra fail to be right proper…)
fail to be right proper…
Just to add: But is close enough to right properness to allow interpretation of the “Motivic Yoga”-fragment, by the property here.
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