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I gave the differential cohomology hexagon its own page (split off from tangent cohesive (infinity,1)-topos), added an Idea-section, added references, and expanded the formal discussion a little bit more.
For those who haven’t seen it yet: there is a ballet choreography dancing an artistinc impression of the differential cohomology diagram here:
started adding Examples-sections, so far there is now
I only saw the 4 min extract. Visually interesting, but are we supposed to be able to see any detailed representation of the elements of the hexagon?
I seem to remember some element chasing being chereographed several years ago.
From that press article I got away with thinking that apparantly in a talk Simons must have brought the hexagon to the board/screen in a way which made a dynamical visual impression even beyond its actual mathematical content and that it is this impression which led to the choreography. But I don’t know anything beyond what is behind these links (which is not much, apart from the performance itself).
But I think in a way it is true that the hexagon expresses a lot of “dynamics” and so I thought it was fun to link to that dance. I gather that “dancing a thesis” has become a common kind of entertainment in some corners of the world, but here it maybe works better than elsewhere.
Here perhaps is the mother of all interpretive dances for scientific topics, the protein synthesis dance. Enjoy!
Thanks, Todd, I didn’t know this.
Meanwhile I have edited at differential cohomology diagram a bit further: added some details of the hexagon for ordinary differential cohomology at Deligne coefficients and started a section on Hopkins-Singer coefficients.
Amazing!
Glad you like it. What specifically do you find amazing?
Started adding some section On differential K-theory.
Has anyone got the hexagon to work for arithmetic Chow groups?
An arithmetic intersection theory will involve three main ingredients: ﬁrst, a geometric intersection theory over the scheme $X$, the geometric part, second, a “reﬁned” intersection theory over $X_{\infty}$, the analytic part, and ﬁnally an interface relating the geometric and the analytic part. (Cohomological arithmetic Chow rings)
Not that I am aware of. Of course this brings us back to our endless discussion: if somebody exhibits cohesion for analytic spaces, then the differential cohomology diagram also follows there…
Maybe Regensburg is the place to go for answers:
Inside the world of smooth extensions of cohomology theories there are a lot of interesting and important foundational open problems…Another aspect of the theory of smooth extensions of cohomology theories is the striking parallelism with its algebraic (and predating) counterpart (Deligne cohomology and its reﬁnements). The arithmetic geometry side has been developed mainly by Gillet and Soule (see e.g. [81]), while the major analytic contributions are due to Bismut. At the moment we see a coincidence of structures, and it would be an interesting project to ﬁnd more substantial bridges between the two ﬁelds.
How does twisted differential cohomology fit in with the hexagon story?
Some rambling thoughts while I should be doing something else. Maybe from here, how about tensoring the stable homotopy type $\hat{E}$ with an element of the relevant Picard ∞-group? Is there a $Pic(Stab(Smooth \infty Grpd))$?
In the ordinary (unsmooth) case, from here
The Picard group $Pic$ of $Spectra$ is easy enough to describe: it is $\mathbb{Z}$, generated by $\mathbb{S}^1$ the 1-sphere.
So is there a smooth version?
Thanks for the question. This is briefly mentioned at tangent cohesive (infinity,1)-topos in the section Twisted cohomology (but I should highlight at also in the entry on the hexagon).
So over there the discussion is generally for any cohesive $\infty$-topos without highlighting the word “differential” much, but, recall, the neat insight here is: the differential cohomology hexagon exhibits every stable homotopy type in a cohesive $\infty$-topos as a differential cohomology theory.
Specifically for $\mathbf{H}$ a cohesive $\infty$-topos and $P \in \mathbf{H}$ a chosen object “of twists”, then consider the cohesive $\infty$-topos $T_P \mathbf{H}$. This is the $\infty$-topos of twisted differential cohomology theories with twists parameterized by $P$.
More specifically one will often want to consider $E\in CRing_\infty(T_\ast \mathbf{H})$ a cohesive $E_\infty$-ring, let $P \coloneqq Pic(E)$ be its Picard $\infty$-groupoid object, consider the canonical $E$-bundle in $T_{Pic(E)} \mathbf{H}$ and regard in any other $T_X \mathbf{H}$ those bundles of spectra obtained by pullbak of this universal one along atwist map $X\to Pic(E)$. This special construction is the evident smooth refinement of constructions that one may currently find in the literature.
Oh yes, we discussed that on a thread last year.
So there’s still a hexagon-like fracturing of twisted differential cohomology, but now within a context $X$?
Yes. One needs to be careful though: the proof of the exactness of the hexagon uses the fiberwise characterization of homotopy pullbacks. Maybe one needs that $X$ has “enough points” to retain this (like $X$ being a manifold) (?). Need to think more about that.
Ah, I am being stupid, it holds in fact generally for all $X \in \mathbf{H}$. By the statement now here .
To check, where you have
As discussed there (and as is discussed below) for $A \in \mathbf{T}H$
you mean $A \in \mathbf{H}$?
Its component on $\hat E\in T Stab(\mathbf{H})$
you mean $Stab(\mathbf{H})$ or $T_\ast \mathbf{H}$ ?
Ah, thanks for catching that. Yes, both were meant to be in $T_\ast \mathbf{H}$. I have fixed it now in the entry.
Coming back to the twisted case touched upon above: I think for instance the standard example of twisted differential K-theory should come out in terms of the tangent topos as follows:
I’ll concentrate on the degree-3 twist. On “ordinary generalized” cohomology (“ordinary” now meaning: no differential refinement yet) this is exhibited by a bundle of spectra
$\array{ KU &\to& KU//B^2 \mathbb{Z} \\ && \downarrow \\ && B^3 \mathbb{Z} \,. }$The rationalization of this is
$\array{ H \mathbb{R}[b,b^{-1}] &\to& H \mathbb{R}[b,b^{-1}]//B^2 \mathbb{R} \\ && \downarrow \\ && B^3 \mathbb{R} \,. }$The standard differential refinement of the twist is given by the homotopy pullback
$\array{ \mathbf{B}^2 U(1)_{conn} &\to& \Omega^3_{cl} \\ \downarrow && \downarrow \\ B^3 \mathbb{Z} &\to& B^3 \mathbb{R} }$so we want a de Rham model for the rationalized twist above which is a bundle of sheaves of spectra over the sheaf $\Omega^3_{cl}$. That should be the sheaf of twisted de Rham complexes as the twisting 3-form $\omega$ is allowed to vary:
$\Omega^\bullet_{tw} := \underset{\omega \in \Omega^3_{cl}}{\sum} ((\Omega \otimes \mathbb{R}[b,b^{-1}])^\bullet, \mathbf{d}_{dR} + b \omega)$So the degree-3-twisted differential version $(KU_{conn}^{tw} \to \mathbf{B}^2 U(1)_{conn})\in T Smooth\infty Grpd$ of $KU \in T_\ast \infty \mathrm{Grpd}$ should be given by a homotopy pullback in $T Smooth \infty Grpd$ of the form
$\array{ \left(KU_{conn}^{tw} \to \mathbf{B}^2U(1)_{conn}\right) &\to& \left( \Omega^\bullet_{tw} \to \Omega^3_{cl}\right) \\ \downarrow && \downarrow \\ \left( KU//B^2 \mathbb{Z} \to B^3 \mathbb{Z}\right) &\to& \left( H\mathbb{R}[b,b^{-1}]//B^2 \mathbb{R} \to B^3 \mathbb{R}\right) } \,.$And I suppose that does come out right.
(Of course my remark about twisted versions of the hexagon above was a bit off the mark: on the one hand of course the hexagon always exists as a diagram, but its exactness depends on $\Pi$ preserving the base “point” of the spectrum objects, whereas here in the twisted case the base space over which we have spectrum bundles actually changes within the hexagon.)
Ah, thanks for catching that. Yes, both were meant to be in $T_\ast \mathbf{H}$.
I wasn’t sure about the first one, because in remark 1, you just say $A$ is a cohesive homotopy type, not that it’s stable.
Will now ponder on the more interesting part of your last comment.
Right, so that “de Rham differential map” $\Pi_{dR} \Omega A \to \flat_{dR} \mathbf{B} A$ exists as soon as $A$ is at least once deloopable. It doesn’t have to be stable.
So by the way, I keep thinking now of adjusting the cohesive notation to make this come out prettier: I defined $\flat_{dR}$ as the homotopy fiber of $\flat \to id$ and $\Pi_{dR}$ as the homotopy cofiber of $id \to \Pi$, and unstably this does make good sense. But since in the hexagon we always have $\flat_{dR} \circ \Sigma$ and $\Pi_{dR} \circ \Omega$ I am thinking about changing the notation to declaring that $\flat_{dR}$ is the homotopy fiber of $\flat \mathbf{B}(-)\to \mathbf{B}(-)$ and dually for $\Pi_{dR}$. That would still match the unstable examples reasonably well. But then all appearances of “$\Omega$” and “$\Sigma$” in the hexagon would disappear and it would look much cleaner.
Re #20, does it work so that if I give you a smooth parameterized spectrum, you can form its hexagon? Or do I need to know twisting information?
I see there’s a comparison map between twisted algebraic and twisted topological K-theory on p. 10 of this.
I wonder if there’s a process like your $\mathcal{K}$, which could generate interesting smooth parameterized spectrum. In the same paper it says (p. 12):
If $\alpha \in H^3(X, \mathbb{Z})_{tors}$, there is also a category $Vect^{\alpha}$ of $\alpha$-twisted ﬁnite dimensional complex vector bundles…
So could there be a smooth version and then a $\mathcal{K}(\mathbf{Vect}^{\alpha})$ as smooth paramterized spectrum mediating between the algebraic and topological K-theories?
does it work so that if I give you a smooth parameterized spectrum, you can form its hexagon? Or do I need to know twisting information?
So let’s distinguish: the hexagonal diagram as such exists for every object (in $\mathbf{H}$ and in $T \mathbf{H}$) which is at least once deloopable, but it is “exact” (is a bunch of interlocking fiber sequences) by necessity only on objects in $T_\ast \mathbf{H}$.
So if we have any object in $T \mathbf{H}$ sitting in its possibly-non-exact hexagon, we may pick a basepoint in its underlying object in $\mathbf{H}$ and restrict to that. This gives a stable object (the fiber of the spectrum bundle over that basepoint) sitting in an exact hexagon. In this sense the original possibly-non-exact-hexagon is identified as a twisted collection of differential cohomology theories without adding further information.
But in special cases we may make this state of affairs more pronounced: as in the example of twisted KU above, typically one takes the base object that parameterizes the twists itself to be a connective spectrum (the Picard $\infty$-groupoid if the typical fiber has $E_\infty$-structure). Then the space of twists itself fits into an exact hexagon, which is then the base of the possibly-non-exact hexagon that in turn is fiberwise exact.
(I should say this more formally, but not right now.)
Thanks for the pointer to arXiv:1104.4654, am looking at it now.
But, yes, the points-to-pieces transform applied to a twisted differential K-theory spectrum in the above sense but built for the fiber $\mathcal{K}(\mathbf{Vect}^{\oplus})$ instead should be a map from twisted algebraic K-theory to twisted topological K-theory. I’ll think about it.
The “the twisted form of algebraic K-theory introduced by Kahn-Levine” is mentioned on p. 4 of Tabuada’s Voevodsky’s mixed motives versus Kontsevich’s noncommutative mixed motives. That seems to be Motives of Azumaya algebra. Tricky stuff.
Thanks, will look into it. Right now I have to run.
Meanwhile, I am preparing some notes along these lines for my talk in Paris next week. A first version is here.
But that’s preliminary. Don’t look at it unless you have energy for preliminary stuff.
Have todash now to catch one more train.
Now I have a more polished set of talk notes on the differential cohomology hexagon at
Differential generalized cohomology in Cohesive homotopy type theory (schreiber)
Do you think it’s the ’certified’ side of HoTT that’s important, or the concept formation side? How big a deal was the case Voevodsky described of an insecure theorem?
Typos: magentic; to large degree that (missing ’be’); GlobOrbitCateory; bounary
Thanks, David.
I’d think that the value of formalization as such, in any of its stages, from “expressed in a system of formal symbols” via “rigorous mathematics” to “fully formal type checked syntactic proofs” is uncontroversial. Proof security is maybe the most obvious advantage.
Personally I find important the “modularity” aspect, by which I mean that the more formalized a set of ideas is, the more it serves communication across community boundaries (but also between parts of a single reasearcher’s mind…), since it provides common ground independent of implicit community knowledge and/or (worse) community consensus. (That is currently the big problem in fundamental physics, that it is fragmented into subcommunities which have lost common ground on which to agree about their joint foundations.)
Of course this second advantage only plays out if the formalization is intelligible and practicable. Otherwise we’d stick with ZFC and be done with it (which is the kind of idea that makes many physicists reject the spirit of formalization). To my mind the point of HoTT is that it is “synthetic homotopy theory”, which means that there is actually a practical chance and not just an in-principle chance to formalize and check work involving homotopy theory – such as $\mathbb{A}^1$-homotopy theory, for instance.
And that, yes, somehow has to do with concept formation. Finding “practical synthetic axioms” means identifying the core concepts of a field, I’d say.
(As in homotopy theory: one might think that homotopy theory is to do with the concept of topological space (its original “point-set”axiomatics) but indeed that’s not actually a core concept at all of homotopy theory, the actual core concept which provides the “synthetic” formulation is that of identity type.)
The fact that I can sometimes understand what’s going on suggesting something’s good about the communication capacity! Wouldn’t it be easier if ABGHR 14 just came out and told us they’re working in the tangent $\infty$-topos?
Typo: Russel (two ’l’s)
Thanks again, fixed now.
The fact that I can sometimes understand what’s going on suggesting something’s good about the communication capacity!
Indeed. The fact that you and me had and have this kind of conversation I regard as a big success. Being able to usefully and constructively communicate deep truths with a computer is a feat, but being able to do the same between two humans whose specialization does not fully overlap is often just as hard.
Wouldn’t it be easier if ABGHR 14 just came out and told us they’re working in the tangent ∞-topos?
I am thankful that they left some observations for the rest of us. ;-)
Wouldn’t it be easier if ABGHR 14 just came out and told us they’re working in the tangent $\infty$-topos?
Sorry, who?
Todd, we’re speaking now about Urs’s Paris talk Differential generalized cohomology in Cohesive homotopy type theory (schreiber). You’ll see there in the bibliography:
[ABGHR 14] M. Ando, A. Blumberg, D. Gepner, M. Hopkins, C. Rezk, An $\infty$-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology, arXiv:1403.4325
Thanks, David.
have exapanded a bit more in the section Examples – de Rham coefficients.
In the notes, what is the reference in footnote 1?
See … for a review.
Had wanted to write an nLab page for this to link to, but now I hadn’t found the time. Will do this soon.
So we get an exact hexagon for any stable object in an $\infty$-topos $A$ which is cohesive over another $B$. What happens if $A$ is infinitesimally cohesive over $B$? Since the points-to-pieces transform is an equivalence, the hexagon degenerates?
(Remind me, is ’differential cohesion’ synonymous with ’infinitesimal cohesion’? differential cohesive (infinity,1)-topos seems to equate them. One thing that confuses in the entries is going between relative notions of cohesion and absolute ones, i.e., relative to $\infty Grpd$.)
Didn’t we once discuss whether there might be a factorisation process, so that for any $A$ cohesive over $B$ it would be possible to factorise it into its largest infinitesimal part? Might there be a general process so that given merely geometric morphisms, one could extract a maximal cohesive part and maximal infinitesimal cohesive part?
Oh, I see. I had changed terminology at some point, and it wasn’t reflected yet in that entry. I have tried to fix that now.
Originally I had said “infinitesimal cohesion” for what I then called “differential cohesion”, because differential cohesion contains infinitesimal objects (those in the kernel of the infinitesimal shape modality, essentially). But then I realized that I better use “infinitesimal cohesion” for the theory consisting entirly just of those infinitesimal objects. Which makes better sense, I guess (I hope), but so I changed terminology at some point.
And you are right, one should think about the hexagon on the infinitesimal context (in the new, better termionology) now. I haven’t yet, though.
Strictly, would it be wrong to say of a $\infty$-topos $A$ that it is differentially cohesive, or even differentially cohesive over another $\infty$-topos $B$, because really you need to specify a $C$ such that $C$ is cohesive over $B$ and $A$ infinitesimally extends $C$?
Regarding the hexagon in an infinitesimal context, if points-to-pieces is an equivalence, does this mean the extreme left and right entries are trivial, so that a stable object is simply a product?
That’s true! the hexagon shows that a stable infinitesimal cohesive type is the product of “its points” as seen by $\flat$ with its differential part as seen by $\flat_{dR}$. In particular if there is a unique global point ( so that the type is a “formal deformation problem” in Lurie’s terminology) then it is $\flat_{dR}$-modal. Yes, good point.
Regarding terminology: yes, strictly speaking “differentially cohesive” involves the choice of two subtoposes, yes.
So in, say, $T \mathbf{H}$ over $\mathbf{H}$, a stable object is a product. Hmm, so what is $Stab(T \mathbf{H})$?
Presumably it has as members things like the product of $0 \to B$ and $A \to \ast$.
Oh, I just see that I missed the above comment. Yes, so one will use $Stab(T \mathbf{H})$ at least implicitly when talking about twisted differential cohomology as cohomology in $T \mathbf{H}$, namely whenever the twists themselves are stable types (which is the case considered “usually” (quotation marks since this hasn’t been considered much yet)).
I have spelled out in a bit of detail the lemma and its proof saying that not only does the $(\Pi \dashv \flat)$-fracture square exhibit any stable cohesive homotopy type as the pullback of a $\flat$-anti-modal type along a map of $\Pi$-modal types into its shape, but that conversely all homotopy pullbacks of this form are $(\Pi \dashv \flat)$-fracture squares.
Does anything interesting happen when hexagon fracturing meets Charles Rezk’s global equivariant cohesion?
I have been wondering about what this gives for equivariant spectra, yes. But I don’t know yet.
So they are stable objects in Rezk’s $Top_{Glo}$? And these coincide with Bohmann’s global orthogonal spectra and/or Schwede’s orthogonal spectra?
So for $G$ a compact Lie group, then a “naive” equivariant G-spectrum is a spectrum object in $(Top_{Glo})_{/\mathbf{B}G}$, yes. The orthogonal spectra serve to model the richer “genuine” G-spectra. But I’d say that if we are being abstractly-minded we want to rather think of those as Mackey functors.
I see Bohmann and Osorno had a paper – Constructing equivariant spectra via categorical Mackey functors – out a few days ago:
We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg–MacLane spectra for Mackey functors and for suspension spectra for finite G-sets.
If Rezk’s cohesive $\infty$-topos only gives us “naive” versions of spectra, do we not yet have a cohesive path to “genuine” local/global spectra?
Thanks for the pointer to Bohmann-Osorno 14! Have added that to Mackey functor.
Regarding your question: I find the terminology “naive spectrum” naive, but that’s how they are called. “Naive G-equivariance” is a perfectly good concept of equivariance. It is less rich than the “genuine G-equivariance”, but that doesn’t mean its not the right thing in the right context. Take for instance the “modular equivariant elliptic cohomology”, officially that’s “naive $SL_2(\mathbb{Z}_n)$-equivariant” for each fixed $n$, but I believe few people would look at that and exclaim “oh how naive!” :-)
Equivariant cohomology theory is a typical field which was historically developed by trial-and-error and proceeding by what looked interesting, and of which a conceptual foundation only appeared rather recently. The terminology “naive/genuine” dates from before this happened.
(I seem to remember that renowned experts voiced similar sentiments on MO at various places, but now I don’t have a link handy )
Mark Hovey’s answer here puts them in perspective:
I can hear you objecting–you must be being too naive–what about complete G-universes? I take the point of view that picking a universe corresponds to picking a model structure on the one God-given category of G-spectra. Picking a smaller universe just means localizing the model structure. So the complete universe is the “initial” one, and every other universe is a localization of the complete universe. The naive universe is the “terminal” one, in the sense that it is a localization of every other universe. There are lots of universes corresponding to model structures in between these.
But it seems there’s even equivocation about what is naive.
Anyway, for local, naive G-spectra, Charles’s paper should contain what’s needed to understand the hexagonal fracture.
Thanks for digging out those links!
The three stages of naivety that Charles Rezk lists I liked to highlight as follows:
Getting into the literature on equivariant cohomology is hard because: 1. what you naively expect to see they don’t even mention, 2. what you come up with being more sophisticated they dismiss as “naive” and 3. what they claim is the real thing you would never have thought is even part of the topic.
:-)
Anyway, as you say, the point is that all these concepts have their place and their role to play.
Doesn’t linear cohesive dependent type theory point us towards the true path?
Yes,we had a discussion about that a while back here, I forget where. The infinity-categorical definition of Mackey functors, is a pull-push through spans. One way to read the equivalence of this with genuinely equivariant spectra is to say that the “motivic” quantization of any theory with certain G-actions will be a spectrum equipped with genuine G-equivariance.
I have taken the liberty of adding here under “Surveys/expositions of this include” pointers to my recent talk notes
I see I was asking 5 years ago in #46
Does anything interesting happen when hexagon fracturing meets Charles Rezk’s global equivariant cohesion?
Now we have singular modalities, should we expect the hexagon diagram with $\lt$ and $\subset$ to deliver useful constructions?
On that theme, how about some equivariant additions to the list of open problems at Some thoughts on the future of modal homotopy type theory?
I am finally working on typing up the story of orbisingular cohesion.
Keep being distracted by M5-brane physics, though. Yesterday I saw how the Perry-Schwarz action for $S^1$-compactified M5-s arises from the supercocycles. But then I got stuck on a detail, so that now I am back to typing orbi-cohesion…
Must be a lot of consequences of Hypothesis H to mop up.
Say one were interested in a synthetic treatment of orbi-cohesion, I guess there would need to be some extra characterisation of the modalities to make them behave as wanted. Is there an overlooked section of The Science of Logic where we find that Hegel had foreseen singularities?
I suppose the axiomatic characterization of orbisingular-cohesion is as follows:
A cohesive adjoint triple of (co)modalities $\lt \dashv \subset \dashv \prec$
such that the leftmost modality is localization
$\lt \;\simeq\; L_{Singularities}$at the set of images of finite connected groupoids under the rightmost modality
$Singularities \;\coloneqq\; \big\{ \prec(\ast\!\!\sslash\! G) \;\vert\; G \in Grp_{fin} \big\}$OK. So is there any synthetic way of picking out $Grp_{fin}$?
There’s a variety of cohesion for each global family, right?
To the extent that this is still thought of as a form of cohesion, are there liftings to an analogue of differential cohesion?
The concept of finite groups is readily available in synthetic homotopy theory: Pointed connected types whose loop group has underlying it a finite set.
I am not considering general global families, but just the family of finite groups, with their discrete cohesion. Part of the point is that this makes things work nicely.
Currently I see no motivation for further extending the singular cohesion.
That’s kind of a “Mathematics made difficult” way to define a finite group…
No, for the present purpose it’s the slick way, since what we need is the “global orbit category” of connected finite groupoids. But either way, it doesn’t matter for what David was asking.
Let’s see if I have this straight. In the plain cohesive case, we’ve observed that some pairs of $(\infty, 1)$-toposes, $\mathbf{H}$ and $\mathbf{H}^{\ast}$, behave as though the latter is ’cohesive’ over the former, often where $\mathbf{H} = \infty Grpd$, but not necessarily. So we seek a synthetic axiomatisation for such pairs, and find it in terms of a triple of (co)modalities.
As with many axiomatisations this doesn’t rule out unexpected models, and indeed in this case someone then spots that a range of models crop up in global equivariant situations. A classic case comes from global spaces for each global family. Further models come from slicing global spaces at ’orbispaces’.
So now one might think to characterise what is special about these cases.
The thought of #62 is to limit ourselves (for the moment?) to the global family of finite groups. Then we have a site named ’Singularities’, and sheaves valued in $\mathbf{H}$ we call $\mathbf{H}_{sing}$. Now we might think to characterise $\mathbf{H}$, $\mathbf{H}_{sing}$ pairs synthetically, and this could go along the lines of #60. And here we look to capture such pairs precisely, since $\mathbf{H}_{sing}$ is being constructed from $\mathbf{H}$, so no unexpected models.
Is all this right?
Then I’m left wondering, is the choice of finite groups largely because orbifolds are what’s foreseen to be useful for physics? In terms of mathematical naturalness, the mathematicians seem interested in the global context of all compact Lie groups, and the orbifold situation with finite groups is just an application, so ’orbifold’ only appears in the Preface of Schwede’s book.
The first bit sounds right to me.
The restriction to finite groups is based on the observation that this makes the proof of Example 2.18 at orbifold cohomology work:
For finite groups $G$ and 0-truncated objects $X \in \mathbf{H}_{/ \ast \sslash G}$ is it the case that the rightmost singular-cohesive modality $\prec$ sends the shape of $X$ to the expected object in $G$-equivariant homotopy theory.
This is because at some stage in this proof we are just homming generic singularities $\ast \sslash K$ into $\infty$-groupoids of plots of $X$, and for that to produce the right formula, identified in Charles’ 2014 note, we need that $K$ is geometrically discrete and the $\infty$-groupoid of plots is a 1-groupoid.
If either of these two assumptions is violated, there are additional homotopy groups picked up from the topology of $K$ and/or the higher homotopy groups of the plots. To get the right formula in this general case one will have to consider all $\infty$-groupoids here as equipped with “atlases”. There is no problem in writing this down, but the result is no longer what is computed by the modality $\prec$.
So for the cohesive theory of equivariant cohomology to work out nicely, the equivariance groups must be discrete (and hence finite if also compact, for purposes of further analysis in equivariant homotoy theory).
Incidentally, that apparent restriction is no actual restiction in all applications, where we are indeed dealing with orbifolds, and this convinces us that this is the right way to go.
One should generally compare what an immense struggle it is for instance for tom Dieck to prove the equivariant Hopf degree theorem in the generality of compact Lie groups, and how much work it is for the reader to see through how elegant the situation actually is in the case of finite groups. Some generalizations may superficially seem but as variations on a theme, but change the character of a subject.
In the present case this should not be so mysterious: If we are dealing with cohesion it should be intuitively clear that something breaks if suddenly we sneak in independent cohesive structure by hand (here: topology on the groups in the family).
If we are dealing with cohesion it should be intuitively clear that something breaks if suddenly we sneak in independent cohesive structure by hand
I see. Thanks for this. Looking back at the question of seeing whether orbi-cohesion can talk to ordinary cohesion here, I see it was in terms of discrete groups.
So you’re writing up an article on orbi-cohesion? It would be interesting to catch up with Charles Rezk.
Was having a little think about #57:
If we have
$\array{ \hat E &\stackrel{}{\longrightarrow}& \flat_{dR} \hat E \\ \downarrow &{}^{(pb)}& \downarrow \\ \Pi(\hat E) &\stackrel{ch_E}{\longrightarrow}& \Pi \flat_{dR} \hat E }$then what is this?
$\array{ \hat E &\stackrel{}{\longrightarrow}& \subset_{dR} \hat E \\ \downarrow &{}^{(pb)}& \downarrow \\ \lt(\hat E) &\stackrel{}{\longrightarrow}& \lt \subset_{dR} \hat E }$1 to 68 of 68