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?

$\array{ \hat E &\stackrel{}{\longrightarrow}& \subset_{dR} \hat E \\ \downarrow &{}^{(pb)}& \downarrow \\ \lt(\hat E) &\stackrel{}{\longrightarrow}& \lt \subset_{dR} \hat E }$ ]]>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.

]]>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).

]]>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.

]]>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.

]]>That’s kind of a “Mathematics made difficult” way to define a *finite* group…

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.

]]>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?

]]>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

$Singularities \;\coloneqq\; \big\{ \prec(\ast\!\!\sslash\! G) \;\vert\; G \in Grp_{fin} \big\}$ ]]>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 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…

]]>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 have taken the liberty of adding here under “Surveys/expositions of this include” pointers to my recent talk notes

]]>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.

]]>Doesn’t linear cohesive dependent type theory point us towards the true path?

]]>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.

]]>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 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 )

]]>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?

]]>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.

]]>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?

]]>I have been wondering about what this gives for equivariant spectra, yes. But I don’t know yet.

]]>Does anything interesting happen when hexagon fracturing meets Charles Rezk’s global equivariant cohesion?

]]>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.

]]>