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created an extemely stubby stub Weiss topology, just to record pointer to that cool fact which Dmitri Pavlov advertised on MO (here).
I have no time to expand on the entry right now. But maybe somebody else here does? Would be worthwhile.
Title :Weiss topology and Goodwillie calculus. Your typing fingers are tripping over themselves today!
Thanks, fixed. Sorry.
I started manifold calculus.
Is it worth a separate page to give the result on that MO page? I wonder if there’s a connection between jet $\infty$-toposes and that tower via Weiss topologies.
Hmm, I seem to have been suggesting something like that back over here. Can’t say I remember that.
I’m not sure if “Weiss topology” is really the best name for the concept. Owen Gwilliam recently told me that he introduced this name in his book with Costello, and Theo Johnson-Freyd then used it in a few of his papers, but as far as I can tell, nobody else apart from us three is using this name.
But I don’t know any papers earlier than Weiss’s that use it, so perhaps this name is accurate, and we do need some name for it…
And now the nLab article Weiss topology is the first hit on Google when one searches for it… (Interesting how the nLab can instantly “hijack” any mathematical term merely by creating a stub article about it.)
Re my question at #4, looking around, we don’t seem to have any clear statement about how jet toposes (n-excisive functor – Examples – Goodwillie n-jets) fit in with the Goodwillie calculus.
At Goodwillie calculus it simply says
$\infty$-Toposes of polynomial $(\infty, 1)$-functors
For each $n$, the collection of polynomial (∞,1)-functors of degree $n$ from bare homotopy types to bare homotopy types is an (infinity,1)-topos, the jet topos.
due to ( Joyal 08, 35.5, with Georg Biedermann) See also at tangent (infinity,1)-category, and Charles Rezk, appears as (Lurie, remark 6.1.1.11).
David, maybe say again what your question is. As the quote you give says, the term “jet $\infty$-topos” $T^{(n)}\mathbf{H}$ is just (new) terminology for the collection of $n$-excisive functors $\infty Grpd_{fin}^{\ast/} \to \mathbf{H}$. These form the $n$th stage of the Goodwillie tower.
These form the $n$th stage of the Goodwillie tower.
So that isn’t explicitly written anywhere (and the quote didn’t even mention a general topos $\mathbf{H}$).
But given that the Goodwillie calculus and (enriched) manifold calculus are examples of a general construction (as in Dimitri’s MO answer), what is to be said generally here?
We’re given some kind of $(\infty, 1)$-functor $F: C \to D$ (with some conditions on these?). There is a tower of approximations. And these can be thought of belonging to the jet toposes of what?
Let me just amplify that the “n-jet toposes”-business is just “terminological sugar” for those categories of $n$-excisive functors out of finite pointed homotopy types. The key theorem is that $n$-excisive reflection is left exact, so that with $\mathbf{H}$ an $\infty$-topos, also $Exc^n(\infty Grpd_{fin}^{\ast/},\mathbf{H})$ is an $\infty$-topos.
I seemed to remember that it was you who had suggested that hence one should call these “$n$-jet $\infty$-toposes” :-)
The entry on n-jet toposes did say this, but I have now highlighted it a bit more there.
Ok, so what I need is the way to tie together ’categories of n-excisive functors out of finite pointed homotopy types’ with the Goodwillie calculus. That entry begins:
The operation of stabilization that sends an (∞,1)-category $C$ to the stable (∞,1)-category $Stab(C)$ does not in general extend to a functor.
We may think of this operation as the analog of linearizing a space. Turning an (∞,1)-functor $F : C \to D$ into a functor $Stab(C) \to Stab(D)$ is not unlike performing a first order Taylor expansion of a function.
This is what Goodwillie calculus studies.
So given nice enough $C$, $Stab(C)$ is arising as excisive functors $Exc(\infty Grpd_{fin}^{\ast/},C)$? And
Turning an (∞,1)-functor $F : C \to D$ into a functor $Stab(C) \to Stab(D)$
is constructing the functor $Exc(\infty Grpd_{fin}^{\ast/},C) \to Exc(\infty Grpd_{fin}^{\ast/},D)$? And higher approximations are maps between higher jet-toposes?
And if that’s on the right track, how to relate this to Dimitri’s
Consider an arbitrary site (or an ∞-site) S. In fact, the constructions below only depend on the underlying topos (or ∞-topos) T of S, and not on S itself. Below “sheaf”, “∞-sheaf”, “stack”, and “∞-stack” are all synonyms for presheaves (of spaces) that satisfy homotopy descent.
The nth Weiss topology (n≥0 or n=∞) on T is defined by declaring a family {U_i→X} to be a covering family if its kth cartesian power {U_i^k→X^k} is a covering family of X^k in T for any 0≤k≤n. If m≤n, then the mth topology contains the nth topology. The category of n-polynomial functors is defined to be the category of sheaves in the nth Weiss topology. The 1st Weiss topology almost coincides with the original topology (for k=0 we see that the empty cover (of the intitial object) is excluded from the 1st Weiss topology), so a sheaf in the ordinary sense is a sheaf in the 1st Weiss topology that is reduced.
Given a presheaf F on T, i.e., a functor T^op→Spaces (one can also take Sets or any other nice target category), we define the nth Taylor approximation T_n(F) as the sheafification of F in the nth Weiss topology. We have a canonical tower F→T_∞(F)→⋯→T_n(F)→⋯→T_0(F).
If S=sSet^op, we recover the homotopy calculus.
It all comes down to just the following fact: for $\mathbf{H}$ an $\infty$-topos (or just a “differentiable (infinity,1)-category”), then there is a sequence of reflective subcategories (the “n-excisive reflections”)
$Exc^0(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow Exc^1(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow Exc^2(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow \cdots \hookrightarrow Func(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \,.$This gives for each object $F$ in $Func(\infty Grpd_{fin}^{\ast/}, \mathbf{H})$ a tower of reflections
$F \to \cdots \to P_3 F \to P_2 F \to P_1 F \to P_0 F \,.$This is the Goodwillie-Taylor tower that exhibits $P_n F$ as being analogous to a degree-$n$ polynomial approximation to the “fuction” $F$. Much of the theory is concerned with worrying about whether and how this converges, i.e. whether $F$ may be reconstructed from its polynomial approximations.
Notice, with regard to previous discussion that we recently had, that when $\mathbf{H}$ indeed is an $\infty$-topos, then
$Func(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq \mathbf{H}[X_\ast]$is the classifying $\mathbf{H}$-topos for pointed objects, the result of going to the internal logic of $\mathbf{H}$ and decreeing
Let there be a pointed object!
In terms of this (pointed) thing-in-itself $X_\ast$ then (by the discussion here)
$T \mathbf{H} \simeq Exc^1(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \simeq \mathbf{H}[X_\ast][(\Sigma\Omega X_\ast^\bullet \to X_\ast^\bullet)^{-1}]$is the result of going to the internal logic of $\mathbf{H}[X_\ast]$ and decreeing
Let all finite pointed powers of that pointed object be linear!
Great. Certainly something to paste in.
And this is a special case of what’s at n-excisive (∞,1)-functor, where it talks about approximating $F$ in $Func(\mathcal{C}, \mathcal{D})$, where $\mathcal{C}$ is an (∞,1)-category with finite (∞,1)-colimits and a terminal object, and $\mathcal{D}$ is a Goodwillie-differentiable (∞,1)-category.
And the latter covers the tower approximations to functors in Manifold calculus and homotopy sheaves:
a topological (or ∞) category C equipped with a Grothendieck topology possessing good covers and, given a presheaf F on C, one can construct the tower of homotopy sheafifications of F - its Taylor Tower - and give an explicit model for it as a tower of homotopical approximations with respect to certain subcategories of C?
Okay, thanks. If you have the energy, could you indeed paste that in where it seems to be missing?
Hmm, I realise there’s still something I’m not seeing.
Given a smooth map $f: M \to N$ between two manifolds, we speak of its $k$-jet.
The Goodwillie analog of $f$ is a functor $F : C \to D$ which preserves filtered colimits (Higher Algebra, p. 755).
The image of $F$ (under left adjoint to inclusion) in $Exc^k(C, D)$ (with some conditions on $C$ and $D$) is $P_k(F)$ (p. 757).
So the questions
I guess I’m wondering if the $J^k$ construction is functorial, so that there’s $J^k(F): J^k(C) \to J^k(D)$, which is related to $P_k(F)$.
I guess 6.2 of Higher Algebra is heading in this direction. So if in corollary 6.2.3.22, the set $S$ is a singleton, then there’s an equivalence between $Exc_{\ast}(Sp(C), Sp(D))$ and $Exc_{\ast}(C, D)$ for suitable $C$ and $D$.
Looking further through 6.2, that makes me feel better. I’m wondering if it can be extended from tangents to jets.
By the way, functors $\infty Grpd^{\ast/} \to \infty Grpd^{\ast/}$ which preserve filtered colimits are equivalent to functors $\infty Grpd^{\ast/}_{fin} \to \infty Grpd^{\ast/}$.
That’s how one translates from the picture of “functions” from $\infty Grpd^{\ast/}$ to itself to elements of the classifying topos $\infty Grpd[X_\ast]$.
As we discussed before, Anel-Finster-Joyal amplify that the analogy to calculus works better in this second picture: think of $\infty Grpd[X_\ast]$ as analogous to a polynomial ring $R[X]$. Then the reflection $P_n$ is projection onto degree-$n$ polynomials.
But in either case, let’s not get the analogy in the way of the maths. When it becomes more cumbersome to carry the analogy along than to just work out the maths, then the analogy is not doing its job.
let’s not get the analogy in the way of the maths
I’m not sure I was doing that. I was just getting a sense that there ought to be a lot more worked out about derivatives, etc. And of course there is. So will Anel-Finster-Joyal cover chap. 6 of Higher Algebra in their terms?
The one place that ’jet’ occurs there is in the sense of ’jet equivalence’:
Definition 6.3.4.1. Let $C$ be an ∞-category which admits finite colimits and has a final object and let $D$ be a differentiable ∞-category. We will say that a natural transformation $\alpha : F \to G$ of functors $F, G : C \to D$ is a jet equivalence if $\alpha$ induces an equivalence $P_n F \to P_n G$ for every integer $n$.
Can that be give a $[X_\ast]$-type formulation? I guess were $C$ pointed spaces, then $\alpha$ induces an equivalence in $D[X_{\ast}]$.
Can that be given a $[X_\ast]$-type formulation?
“An $n$-jet equivalence between two polynomials is a map between them that is an equivalence up to degree $n$.”
But what I meant re analogy is that this is just language. You seem to be after whether it’s “really jets” or “really polynomials”. What I am trying to say is that to some extent the maths here is evident enough in itself (the Goodwillie-Taylor tower) that when we find ourselves spending more time worrying about whether $P_n$ is to be called an “$n$-jet projection” or a “degree $n$-polynomial projection” than we need to build $P_n$ itself, then maybe it’s not so important.
But maybe I am missing what you are after.
I’m not sure I know what I’m after. What’s utterly clear is that’s plenty missing at the nLab in this area. So when a new observation comes along, it’s a good opportunity to spur us on to bring things together.
E.g., how does the Weiss topology story at MO fit with the $Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H})$ story?
E.g., how does the Weiss topology story at MO fit with the $Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H})$ story?
It provides sites of definition.
So we know abstractly from the n-excisive reflection theorem that for $\mathbf{H}$ an $\infty$-topos, then
$Exc^0(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow \cdots \hookrightarrow Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow Exc^{n+1}(\infty Grpd_{fin}^{\ast/}, \mathbf{H}) \hookrightarrow \cdots \hookrightarrow \mathbf{H}[X_\ast]$is a sequence of sub-$\infty$-topos inclusions. (Incidentally that means that $\mathbf{H}[X_\ast]$ carries a sequence of modalities $P_0 \lt P_1 \lt P_2 \lt P_3 \lt \cdots \lt id$.)
Now given that the $\mathbf{H}$-topos $\mathbf{H}[X_\ast]$ has as site of definition $(\infty Grpd_{fin}^{\ast/})^{op}$ with its trivial Grothendieck topology, we might want to know the Grothendieck topology on $(\infty Grpd_{fin}^{\ast/})^{op}$ whose $\infty$-sheaf $\infty$-topos is $Exc^n(\infty Grpd_{fin}^{\ast/}, \mathbf{H})$.
The answer is: it’s the $n$th Weiss topology for $(\infty Grpd_{fin}^{\ast/})^{op}$.
The answer is: it’s the $n$th Weiss topology for $(\infty Grpd_{fin}^{\ast/})^{op}$.
What’s the original topology on $(\infty Grpd_{fin}^{\ast/})^{op}$ in this statement? If you start with a site with trivial topology, it seems to me that the $n$th Weiss topology is also trivial for all $n\geq 1$.
[never mind]
Ah, but according to Jacob Lurie’s comment here, every $n$-excisive functor is a sheaf for the topology where all nonempty sieves are covering. The only way to enlarge this topology is to declare that some objects are covered by the empty sieve. But then a $0$-excisive functor with non-contractible value is not a sheaf anymore, so the localizations involved in the Goodwillie tower cannot be topological.
Maybe Dmitri could expand, he makes the statement in question in the MO comment (here) that started the thread. Checking in Boavida-Weiss, they don’t actually seem to deal with the situation of Goodwillie calculus, though.
I might have misunderstood something, but Jacob’s comment in http://mathoverflow.net/questions/215771/a-universally-non-hypercomplete-infty-topos is about inverting Čech covers associated to a single morphism. As far as I can see, he says nothing about families of morphisms, let alone sieves. Indeed, the very point of Weiss topologies is that they are nonsuperextensive, and there is a huge difference between singleton families and arbitrary families. (This is unlike most classical situations, which are superextensive.)
Let me illustrate this point for excisive functors. An excisive functor sSet→sSet sends homotopy pushout squares to homotopy pullback squares. Equivalently (reversing arrows), an excisive functor is an ∞-presheaf on sSet^op such that F(X) → F(U) ×_{F(U ×_X V)} F(V) is an equivalence for any homotopy pullback square in sSet^op with arrows U→X, V→X, U ×_X V → U, U ×_X V → V (this corresponds to a homotopy pushout square in sSet with arrows X→U, X→V, U → U ⊔_X V, V → U ⊔_X V).
Thus the descent condition in this case is nontrivial, even if the condition pointed out by Jacob is always satisfied.
I might have misunderstood something, but Jacob’s comment in http://mathoverflow.net/questions/215771/a-universally-non-hypercomplete-infty-topos is about inverting Čech covers associated to a single morphism. As far as I can see, he says nothing about families of morphisms, let alone sieves.
It makes no difference: the Grothendieck topology generated by singleton covers contains all nonempty sieves. Given a collection of coverings stable under base change, Cech descent for these coverings is equivalent to descent for the topology they generate.
My point is that the descent condition you spelled out cannot be descent with respect to a Grothendieck topology, even though it defines a left exact localization.
Given a collection of coverings stable under base change, Cech descent for these coverings is equivalent to descent for the topology they generate.
True, but not every Grothendieck topology is generated by its singleton covering families, and the Weiss topology certainly isn’t, so I don’t see a contradication: for a Grothendieck topology T we can have a presheaf that satisfies descent with respect to singleton covering families (and therefore the Grothendieck topology generated by singleton covering families), yet this presheaf may fail to satisfy descent with respect to T.
In the terminology of the article superextensive site, the topology T_cov generated by singleton covering families may be strictly smaller than T itself. For superextensive sites T is generated by T_cov and the covering of disjoint unions by their components; but Weiss sites are not superextensive.
The contradiction is that Jacob’s topology is the largest topology for which n-excisive functors are sheaves. So there is no room for Weiss topologies. In fact, there cannot be any coverage such that Cech descent is equivalent to n-excisiveness, for then n-excisive functors would be sheaves for the induced topology.
[Here I mean coverage in the weakest possible sense: just a collection of coverings with no conditions.]
The sheaf condition says that the map F(X)→F(U) ×_{F(W)} F(V) is an equivalence.
That’s not what the sheaf condition says, you’re ignoring self-intersections.
Also the cover you’re adding was already in Jacob’s topology. Any nonempty cover is.
and the descent object for covering families of cardinality 2 can be computed as the homotopy pullback.
Consider for example a cover of the form $\{X\amalg X\to X,\emptyset \to X\}$ (eg on the site of manifolds).
In fact, I don’t think the question of how the descent condition really looks like is even relevant for the discussion; simply taking the left exact localization with respect to the morphism of presheaves (on sSet^op) of the form Y(U) ⊔_Y(W) Y(V) → Y(X) enforces the property of 1-excisiveness.
Re #32: Yes, this cover is not a basal cover (see Verdier site): the diagonal map U→U ×_X U is not a covering map when U=X⊔X.
The covers in the topology on sSet^op under consideration are basal covers, as I somewhat clumsily tried to explain in #31.
For such a cover we can replace self-intersections U ×_X U in the resulting diagram by U, then throw away duplicates, and obtain a cubical diagram of the desired shape, in particular, for covering families with two elements we get a pullback diagram.
I’m not sure what point you’re trying to make. You’ve agreed earlier in #27 that if a presheaf has Cech descent for all singleton families then it has Cech descent for all nonempty families. So you must agree that the only way to enlarge Jacob’s topology is by throwing in empty covers.
You’ve agreed earlier in #27 that if a presheaf has Cech descent for all singleton families then it has Cech descent for all nonempty families.
Only for a superextensive site; the Weiss site is not superextensive.
No, it is true for any site with pullbacks. The proof is easy: let U be a nonempty sieve on X; pick any morphism in it and let V be the generated sieve. Then we have V ⊂ U ⊂ Hom(-,X). The composite inclusion becomes an iso when you sheafify, hence so does the second inclusion.
Re #37: Yes, I see what you mean now: all maps form singleton covers, therefore all nonempty sieves are covering.
So the “sieves” that I want to invert are no longer ∞-monomorphisms into representables, but rather some arbitrary maps. (I guess this means we have here a nontrivial example of a “Grothendieck hypertopology” discussed in https://nforum.ncatlab.org/discussion/6855/grothendieck-hypertopologies-model-all-toposes/.)
Yes. In fact, the whole Goodwillie tower must consist of purely cotopological localizations of the classifying ∞-topos for pointed ∞-connective objects. Replacing $U\times_X U$ by $U$ as you suggest in #31, gives a 1-coskeletal hypercover, but there are again self-intersections like $U\times_X U\times_X V$ in degree 2. Presumably, if you keep covering those by diagonal maps, you will get a hypercover for which descent is equivalent to the 1-excisiveness condition.
If someone could write up the gist of this somewhere appropriate on nLab, that would be great. I see the ’cotopological localization’ just above hasn’t been created yet.
With a spare moment this morning I got cotopological localization started. There are still things to be added, links to be made, etc. E.g, at hypercompletion, should its characterization as maximal cotopological localization appear in the Idea section, or later?
Re #39: Yes, precisely, replacing all self-intersections of U with U does produce a hypercover that gives the right answer, but this hypercover cannot be refined by an ordinary Čech cover.
Sorry to dredge this back up.
It seems a conclusion about homotopy calculus was reached in #39 and #42. I’m having trouble following, but surely the conclusion is the following. An $n$-excisive functor $F: Top \to Top$ is one such that for every $A \in Top$ and every $A_0, \dots, A_n \in Top^{A/}$, $F$ satisfies descent with respect to the co-hypersieve $U$ of $A$ whose $k$th level consists of the coproducts (in $Top^{A/}$) of $k+1$-element subsets of $\{A_0,\dots,A_n\}$. So much is a rephrasing of the usual definition. Call the hypertopology on $Top^{op}$ for $n$-excisive functors $J_n$.
What I don’t see from this is how to view the hypertopology for $n$-excisive functors as the $n$th “Weiss hypertopology” derived from the hypertopology for $1$-excisive functors, which I thought was the original claim. I understand this to mean the following. If $J$ is a hypertopology, then the $n$th Weiss hypertopology $W_n(J)$ consists of the “$n$-power $J$-hypercovers”. That is, a hypersive $U \to X$ is in $W_n(J)$ if and only if $U^k \to X^k$ is in $J$ for every $1 \leq k \leq n$.
I understand the original claim from Dmitri Pavlov’s MO post to be that $J_n = W_n(J_1)$, but I don’t understand at all why this is so.
Re #43: For the homotopy calculus we must change the way we produce morphisms of presheaves at which we localize.
For the manifold calculus, given a covering family in Man (in the nth Weiss topology), we produce a (Čech) hypercover by taking iterated fiber products. We then invert these hypercovers in the category of ∞-presheaves, which gives us the desired category of ∞-sheaves that recovers the manifold calculus.
For the homotopy calculus, given a covering family in sSet^op (in the nth Weiss topology), we produce a hypercover by taking iterated fiber products, replacing any self-intersection such as U ×_X U with U. We then invert these hypercovers in the category of ∞-presheaves, which gives us the desired category of ∞-sheaves that recovers the homotopy calculus.
Re # 44: Okay, so it’s a multistep process:
Start with an actual topology $J$ (not a hypertopology) on a category $C$.
Derive from $J$ the $n$th Weiss topology $W_n(J)$: A cover in $W_n(J)$ is a collection of maps $\{U_i \to X\}_i$ such that $\{U_i^k \to X^k\}_i$ is a $J$-cover for all $1 \leq k\leq n$.
Use two different procedures to derive a hypertoplogy from $W_n(J)$.
Localize presheaves on $C$ with respect to the hypertopology.
So I guess my confusion is the same as Marc’s in #21: What is the original topology $J$ on $Spaces^{op}$ from which this procedure produces the category of $n$-excisive functors? A number of possibilities were raised, but I’m not sure what the conclusion was, especially since most of that discussion took place before the need for step (3) was recognized.
Re #45: Covering families in Spaces^op are nonempty sieves. For instance, for a covering family of cardinality 2 the descent condition will require a certain diagram to be a homotopy limit diagram. After removing redundant terms, we arrive at the situation described in #29.
Re #46: But then I have the same issue that Marc had in #21: if $J$ consists of all nonempty covers, then $W_n(J)$ also consists of all nonempty covers (at least as I understand $W_n(J)$ to be defined, as in step (2) of #45), so the $n$th Weiss topology is the same as the original topology: $J = W_n(J)$. Maybe this is fixed by doing different things in step (3) of #45, but then it seems that the Weiss topology (step (2) of #45) is playing no role in the homotopy calculus.
The cardinality of the cover does seem relevant to defining the correct localization for $n$-excisive functors (i.e. $n$-excisive functors satisfy descent with respect to hypercovers coming somehow from $n+1$-element covers), but I don’t see how the cardinality of the cover comes up in the steps outlined in #45.
I think I agree that there is a uniform way to turn all nonempty covers into hypercovers such that localizing at these hypercovers is the 0-excisive localization. For $n$-excisive localization, it seems to me that the thing to do is to restrict to covers with at least $n+1$ elements. But I don’t see what this has to do with the Weiss topology.
Maybe the thing to do is to give up on step (2) of #45 being the same for the manifold calculus as it is for the homotopy calculus. In the manifold calculus, we restrict to $n$-Weiss covers, while in the homotopy calculus, we restrict to covers of cardinality $\geq n+1$.
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