It seems that, over 4 years on, little has been made of Charles’s preprint. Only 6 hits on Google Scholar, 3 due to the same person.

]]>To come back to the (maybe vain) issue of how to think of the cohesion of the global homotopy theory along the lines of what it says at motivation for cohesive toposes:

A “cohesive blob” to be thought of as analogous to a drop of water molecules held together by cohesion is whatever $\Pi$ sends to the point. Now what $\Pi$ sends to the point are foremost automorphisms. We might think of an elementary such cohesive blob as a bunch of copies of an “actual” point $x$ that are all held together by automorphisms:

$\left( \array{ x &\stackrel{\simeq}{\to}& x \\ & _{\mathllap{\sim}}\searrow & \cdots \\ && x &\stackrel{\simeq}{\to}& x \\ && && \downarrow^{\mathrlap{\simeq}} \\ &&&& x } \right)$Just a thought. Maybe that’s a pointless comment, though.

]]>Thanks, Charles, for the comments. I certainly see what you are getting at.

What I had vaguely been thinking of was maybe different. The fact that the $(\Gamma \dashv \nabla)$ bit that you find on $Top_{Orb}$ is roughly (roughly) like the $(\Pi \dashv \Delta)$-bit on $ETop\infty Grpd$ vaguely reminded me of differential cohesion, which is cohesion extended with more adjoints that do not quite make a longer total sequence of adjoints, but do “with a jump included”: after passing to the corresponding (co-)monads the structure for differential cohesion is this:

$\array{ && && Red \\ && && \bot \\ && \int & \subset & \int_{inf} \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & \flat_{inf} \\ \bot & & \bot && \\ \ast & \subset& \sharp }$Here in the middle we have the familiar shape modality, flat modality, sharp modality and then on the right further: reduction modality, infinitesimal shape modality, infinitesimal flat modality. The inclusion sign means inclusion of subcategories of “modal types”.

So this is a situation where the sequence of adjoints sort of continues one step further, but only after some re-adjustment of sorts.

I have really no precise idea if this has anything to do with what you see in global homotopy theory, I was just being vaguely reminded of it. I thought maybe there might be a sense in which objects in $Top_{Orb}$ are a bit like “infinitesimally thickened” objects of $ETop\infty Grpd$. (Not sure, don’t read much in this last sentence, it’s just a shot into the blue, for lack of time to really think about it in detail.)

]]>One more point.

Urs, perhaps you are hoping for some kind of triangle of cohesion, relating the $\infty$-toposes $\mathrm{Top}$, $\mathrm{Top}_{\mathrm{Orb}}$, and $\mathrm{ETop}\infty\mathrm{Grpd}$. This doesn’t really work well.

I think the right picture is a square of cohesions:

$\array{ \mathrm{ETop}\infty\mathrm{Grpd} & \to & \mathrm{OTop}\infty\mathrm{Grpd} \\ \uparrow & & \uparrow \\ \mathrm{Top} & \to & \mathrm{Top}_{\mathrm{Orb}} }$In the upper right corner, I used “O” for orbifold. Roughly, I am imagining this upper right corner to be an $\infty$-topos of sheaves on orbifolds (locally modelled by $M//G$ where $G$ is a compact Lie group), with respect to a *suitable* topology.

There is an *unsuitable* topology, whose sheaves will just be $\mathrm{ETop}\infty\mathrm{Grpd}$ again. Covers $U\to X$ for the “suitable topology” will probably be “unsuitable covers” which satisfy an additional condition: namely, that they are surjective on $G$-isotropy for all compact $G$ (i.e., all $*//G\to X$ admit a lift to a map $*//G \to U$). The distinction between the two topologies is analogous to that between etale and Nisnevich topologies in motivic homotopy theory, and is what I was trying to allude to in comment 25 above.

To continue with the analogy to the motivic homotopy theory story: a *cohomology theory* on $\mathrm{ETop}\infty\mathrm{Grpd}$ should be a functor $F:\mathrm{ETop}\infty\mathrm{Grpd}^{\mathrm{op}}\to \mathrm{Spectra}$ taking colimits to limits, and such that $F(X\times A^1)\approx F(X)$ for all $X$, where $A^1$ is the real line as a topological manifold. Such functors must factor through the “$A^1$-localization” of $\mathrm{ETop}\infty\mathrm{Gpd}$ obtained by formally inverting $X\times A^1\to X$ for all $X$; if you work this out, I think the $A^1$-localization of $\mathrm{ETop}\infty\mathrm{Grpd}$ turns out to be equivalent to $\mathrm{Top}$, and in fact you discover that any such cohomology theory on $\mathrm{ETop}\infty\mathrm{Grpd}$ satisfies $F(X)\approx F(\Delta \Pi X )$, where $\Pi\colon \mathrm{ETop}\infty\mathrm{Grpd}\rightleftarrows \mathrm{Top}:\; \Delta$. Thus, cohomology theories on $\mathrm{ETop}\infty\mathrm{Grpd}$ in this sense are just the usual ones.

In particular, equivariant cohomology theories such as equivariant $K$-theories don’t define cohomology theories on $\mathrm{ETop}\infty\mathrm{Grpd}$. I would expect that the $A^1$-localization of $\mathrm{OTop}\infty\mathrm{Grpd}$ should be $\mathrm{Top}_{\mathrm{Orb}}$, and thus interesting equivariant cohomology theories can be defined on $\mathrm{OTop}\infty\mathrm{Grpd}$.

]]>Urs: no, $\nabla$ cannot have a right adjoint.

I started to reply to your comment, and ended up rambling. My ramble is appended below, though I don’t know if it helps.

Here is an observation: although the object $\mathbf{B}G$ in $\mathrm{Top}_\mathrm{Orb}$ is a kind of a classifying object, it does not classify what you might expect.

In $\mathrm{Top}$, consider $*\to BG$, and build the “Cech complex” on this map. This is a simplicial object $[n]\mapsto G^n$, whose realization $|G^n|\approx BG$ recovers $BG$. This is closely tied to the fact that $BG$ classifies $G$-torsors in $\mathrm{Top}$ (in the $\infty$-topos sense).

Likewise, in $\mathrm{ETop}\infty\mathrm{Grpd}$, consider $*\to \mathbf{B}G$ for a Lie group $G$. The Cech complex is $[n]\mapsto G^n$ (where $G$ is the image of the manifold in $\mathrm{ETop}\infty\mathrm{Gpd}$), and the realization of it is $\mathbf{B}G$. Thus, $\mathbf{B}G$ actually classifies principal $G$-bundles.

In $\mathrm{Top}_{\mathrm{Orb}}$, this fails. The Cech complex of $\mathbf{B}G$ is $[n]\mapsto \Delta(G^n)$, whose realization is $\Delta(BG) \approx \Delta\Gamma(\mathbf{B}G)$ (in the language of $\mathrm{Top}_{\mathrm{Orb}}$), which is not the same as $\mathbf{B}G$. In other words, $\mathbf{B}G$ is not the naive classifying object for $G$-torsors in $\mathrm{Top}_\mathrm{Orb}$.

It *is* a kind of classifying object, not for $G$-torsors, but for “equivariant principal $G$-bundles”. For instance, if $H$ is another compact Lie group acting on a nice $H$-space $X$, then (in $\mathrm{Top}_{\mathrm{Orb}}$) we get an object $\delta_H(X)$ (the image of $X$ under the composite $H\mathrm{Top} \xrightarrow{\Delta_H} \mathrm{Top}_{\mathrm{Orb}}/\mathbf{B}H \to\mathrm{Top}_{\mathrm{Orb}}$). It turns out that

the space of $H$-equivariant principal $G$-bundles over the $H$-space $X$. (This is hidden in section 3.5 in my note; I should make it more prominent.) These are not simply $G$-torsors over $\delta_H(X)$ in $\mathrm{Top}_{\mathrm{Orb}}$.

This will probably still seem confusing: after all, in $\mathrm{ETop}\infty\mathrm{Grpd}$, for $H\curvearrowright M$ (a Lie group acting on a manifold), the space $\Map(M//H, \mathbf{B}G)$ really is a space of $H$-equivariant principal $G$-bundles on $M$. So, in the $\infty$-topos $\mathrm{ETop}\infty\mathrm{Grpd}$, the notion of “$G$-torsor” apparently includes these fancy equivariant principal $G$-bundles.

However, this does not work so well in equivariant homotopy $\infty$-toposes, such as $\mathrm{Top}_{\mathrm{Orb}}$ (or for that matter, $H\mathrm{Top}$, as the same distinction between “naive classifying space” and “equivariant classifying space” of $G$ is present there).

]]>Charles,

one reason why it may be hard to match the intended intuition for cohesion to the cohesion you find is that, to my mind at least, the cohesion you find is curiously “shifted to the left” in an unexpected way.

Here is what I mean.

So my intuition is — we talked about that above around #10 — that $Top_{Orb}$ is similar to Euclidean-topological infinity-groupoids, the latter being a bit bigger. That’s what the Gepner-Henriques result suggests, I suppose, but I may be missing something here.

Now also $ETop \infty Grpd$ is cohesive over $Top$ (I am sticking to the notation of your note now, usually I would write $L_{whe} Top$ or $\infty Grpd$ instead), of course. But the functor $Top \to ETop \infty Grpd$ which is like (under the above intuition) your $\nabla$ on your p. 12 is instead the $\Delta$ of $E Top \infty Grpd$.

Here I mean is that the functor $F : Top \to ETop \infty Grpd$ with the property that for $G$ any compact Lie group (and $\mathbf{B}G = \ast //G \in ETop \infty Grpd$ its topological quotient stack) we have

$Map_{ETop \infty Grpd}(\mathbf{B}G, F(T)) \simeq Map_{Top}(B G, T)$naturally for $T \in Top$, that this functor $F$ is the $\Delta$ of $E Top \infty Grpd$, not the $\nabla$, as it is for $Top_{Orb}$, according to your discussion on p. 12.

And related to this “shift to the left” of the adjoint quadruple is the other one: if $X \in ETop \infty Grpd$ is equipped with a $G$ infinity-action, exhibited by the object $(X//G \to \mathbf{B}G)$ in the slice $ETop \infty Grpd_{/\mathbf{B}G}$, then what takes this action to its homotopy quotient is the $\Pi_{/\mathbf{B}G}$ of $ETop\infty Grpd$. But for $Top_{Orb}$ this is instead what $\Gamma$ does, according to your p. 12 again.

So for this reason IF we think of objects of $Top_{Orb}$ as roughly being topological stacks (orbispaces) then your $\Gamma$ is very much unlike what my “motivation for cohesion” asks $\Gamma$ to be, because it is instead very much like what my “motivation for cohesion” asks $\Pi$ to be.

That’s what makes the intuitive interpretation here confusing, to me at least.

This motivates the following question, which however I haven’t thought about at all on a technical level, it’s just motivated by analogy for the moment:

- Is there any chance that your $\nabla$ might have further
*right*adjoints?

David: It is not settled, but I do not plan to do anything particularly fancy; merely survey work related to the power operations stuff I’ve been working on.

]]>Urs:

About motivation for cohesive toposes: I can understand why these particular structures are good things to have, from the point of view of geometry, but I really have no sense of why these are the only kinds of structure to pick out. And in any case, in the homotopical examples I’m thinking of, none of that helps at all. It is mysterious to me, why this particular structure (cohesion) should appear.

About equivariance: I don’t know what to say. The role of field theory is mysterious to me. I’ll note that for anything you want to do with equivariant elliptic cohomology, you may be happy to restrict to abelian Lie groups (since the general theory for arbitrary Lie groups is constructed formally from the abelian case).

Higher equivariance should be captured by “higher global spaces”, where the indexing category (of compact (abelian) Lie groups) in extended to include things like $K(Z,n)$. Or so I would guess.

]]>Incidentally, Charles, can you give us a hint as to your talk topic for the ICM later this year? Or is it not yet settled and/or secret?

]]>regarding the motivational section 1.3 “Equivariant cohomology”:

maybe I may use this as an occasion to ask you for a comment on the following. This may superficially be a bit off tangent regarding the thrust of your text, but let me try anyway (and I am hoping it might actually have some good relation).

You may remember a while back we had a super-brief exchange on equivariant elliptic cohomology. I was trying to get a better feel for why the definitions here are what they are. I was and am discussing this with Joost Nuiten, and at some point the following aspect very much helped me/us, coming, as we were, from a QFT point of view.

Namely we meditated over section 5.1 “2-Equivariant elliptic cohomology” in Jacob Lurie’s “A Survey of Elliptic Cohomology”. There it says we are to allow equivariant elliptic cohomology tobe defined not just for group actions but also for higher group actions, such that we may apply it to $\mathbf{B}^2 U(1)$ regarded as a space with a $G$-$\infty$-action such that its homotopy quotient is the delooping of $G$’s String group, hence to this diagram here

$\array{ \mathbf{B}^2 U(1) &\to& \mathbf{B}String(G) \simeq \mathbf{B}^2U(1) // G \\ && \downarrow \\ && \mathbf{B}G }$and such that the $G$-equivariant ellitptic cohomology of the point is then understood as the space of global sections of the line bundle on the moduli space of $G$-connections on the elliptic curve, which is obtained from this 3-bundle.

This is a statement which I can very much relate to, because this is what we described for instance in *A higher stacky perspective on Chern-Simons theory (schreiber)*. Namely $\mathbf{B}String \to \mathbf{B}G$ may be thought of as the “prequantum 3-bundle” of the WZW model which may be used to give a quantization of the WZW model (in fact of $G$-Chern-Simons theory) as an “extended” TQFT. But if we are just interested in a traditional Atiyah-style QFT then we transgress this 3-bundle to a 1-bundle by homming our wordlsheet (the elliptic curve) into everything and then fiber integrating to get the traditional *Hitchin connection* on the moduli space of $G$-connections on that surface, which is what Jacob Lurie is alluding to in section 5 of his survey.

So this made me start to get a glimpse of a more field theoretic idea of why one wants to be looking at equivariant cohomology as functors on $G$-spaces: somehow we are to think of each such $G$-space as encoding a “prequantum 3-bundle” and we imagine that after transgressing this to a moduli space of gauge fields we form a suitably “linearized” space of sections of that, as in geometric quantization, and hence get an abelian group.

Now, that perspective would immediately make sense (to me) if I restricted attention to $G$-spaces of the “linear” form $\mathbf{B}^n U(1)$ or similar. But from this perspective it still seems mysterious why it makes sense to extend this to all (and hence to “non-linear”) $G$-spaces.

Hm, maybe this is a bad way to ask my question to you. Maybe all I am asking is: do you have any thoughts on how section 5.1 of Jacob Lurie’s elliptic survey might relate to the global equivariant homotopy theory the way you are discussing it?

]]>Nice!

Am beginning to look at it. I am starting a category:reference entry *Global Homotopy Theory and Cohesion* to record this (nothing much in that entry yet).

on page 3 a curly $\mathcal{G}$ appears, which is probably just meant to be a $G$

footnote 6 on p 11 reads: “See Sch14. I don’t actually understand this motivation”

Hah! :-) So, let’s see, is it just that in the nLab entry *motivation for cohesive toposes* I am expressing myself in a maybe overly convoluted way, or do you see what I am trying to say and just don’t agree that it is a decent motivation? It’s meant to be a rather simple idea. If I am failing to convey it, maybe somebody else here has an idea how to do it?

On the other hand, I suppose one issue here is that the cohesion on the global homotopy category that you describe is indeed a bit exotic in as far as the geometric intuition is concerned. Possibly the intuition which the nLab entry means to convey is not actually very useful here. But let me think about it, maybe we just need to find the right perspective…

]]>I put a working document discussing cohesion in global homotopy theory on my homepage.

]]>Interesting. With some people, including notably Thomas Nikolaus, we were also thinking about adapting equivariant K-theory to $E$.

We are looking for good smooth refinements of $KU \in Spectra$ to $\mathbf{KU} \in Spectra(E)$ such that $\Pi(\mathbf{KU}) \simeq KU$.

One sugestive choice is to do the smooth version of the construction appearing in Snaith’s theorem, hence first form the free $E_\infty$-ring object

$\mathbb{S}[\mathbf{B}U(1)] \in E_\infty(E)$from the group stack $\mathbf{B}U(1)$, and then invert the smooth Hopf bundle $\mathbf{\beta}$. Since forming $\mathbb{S}[] = \Sigma_+^\infty$ and inverting elements are $\infty$-colimits which are preserved by $\Pi$, it follows that setting

$\mathbf{KU} \coloneqq \mathbb{S}(\mathbf{B}U(1))[\mathbf{\beta}^{-1}]$satisfies $\Pi(\mathbf{KU}) \simeq KU$.

I am currently trying to see as a first check for $G$ a compact Lie group, if then

$\Pi([\mathbf{B}G, \mathbf{KU}]) \in Spectra$is at all close to the operator $K$-theory spectrum of the reduced $C^\ast$-algebra of $G$.

Hm, now

$[-, \mathbf{KU}] : E^{op} \longrightarrow Spectra(E)$is guaranteed to satisfy descent. But

$\Pi [-, \mathbf{KU}] : E^{op} \longrightarrow Spectra$is not.

That’s the kind of thoughts I keep having. I haven’t thought about this from the perspective of $Glob$ much at all yet.

]]>I think I can explain a little bit better what I’m after here, expanding on my comment 25.

Start with $E=$ homotopy theory of sheaves of $\infty$-groupoids on $\mathrm{Man}$, the category of smooth manifolds. (Or topological manifolds; it really doesn’t matter.) $\mathrm{Man}$ is fully faithful in $E$; more generally, $E$ contains a full subtheory $\mathrm{LieGpd}$ equivalent to that of Lie groupoids. For instance, an action $G\curvearrowright M$ of a Lie group on a manifold gives an object $M//G$ in $E$.

I’m interested in cohomology theories on $E$ which extend a given cohomology theory on $\mathrm{Man}$. These will be functors $F\colon E^{\mathrm{op}}\to \mathrm{Spectra}$ taking homotopy colimits in $E$ to homotopy limits.

Clearly, such $F$ are determined by the restriction $F|\mathrm{Man}$, which is assumed to be a cohomology theory in the classical sense. In particular, $F|\mathrm{Man}$ is homotopy invariant. It is then not hard to see, for instance, that for objects of the form $M//G$ in $E$ we have

$F(M//G) \approx F(M_{hG})=F(EG\times_G M),$the Borel-equivariant $F$-cohomology of $G\curvearrowright M$.

I’m interested in equivariant cohomology theories which are not Borel-type; e.g., equivariant $K$-theory. Equivariant $K$-theory is not a cohomology theory on $E$. :-(

(This may seem a little surprising when you remember thatt $\mathrm{Vect}\colon E^{\mathrm{op}}\to \mathrm{Top}$, which associates to $M$ the $\infty$-groupoid of vector bundles over $M$ (morphisms between vector bundles are the space of gauge transformations, viewed as a homotopy type), is represented by an object of $E$. In fact, $\mathrm{Vect}=\coprod \mathbb{B}U(n)$. But that’s the way it is. The failure of $K$-theory to have the same descent properties as $\mathrm{Vect}$ is familiar in the algebraic geometry setting, where one says that $K$-theory does not have “etale descent”.)

So the proposal above was to define an equivariant cohomology theory $F$ (such as $K$-theory) on $E$, using the functor

$\mathrm{GlobType}\colon E \to \mathrm{Glob}=\mathrm{Psh}(\mathrm{Orb}),$the functor described in comment 11 above, which sends $X$ in $E$ to the presheaf $G\mapsto \Pi[\mathbb{B}G, x]$.

Then $F$ will factor through a cohomology theory $F'\colon \mathrm{Glob}^{\mathrm{op}}\to \mathrm{Spectra}$.

Unfortunately, $\mathrm{GlobType}$ won’t preserve colimits: $F$ isn’t acutally a cohomology theory on $E$. Sad.

It seems like the right thing to do is change $E$. Thus, we should have a *global* version $E$, perhaps modelled as

sheaves of $\infty$-groupoids on a suitable $\infty$-category of (proper) Lie groupoids, under a suitable “Nisnevich-type” topology on $\mathrm{LieGpd}$. Then equivariant $K$-theory should factor

$E_{\mathrm{Glob}}^{\mathrm{op}} \to \mathrm{Glob}^{\mathrm{op}} \to \mathrm{Spectra},$where both functors respect colimits in the right way; in fact, $E_{\mathrm{Glob}}\to \mathrm{Glob}$ should be part of a cohesion.

]]>We could probably consider non-compact groups if we restricted to *proper* actions. And generally, one can, when faced with some theoretical setup involving actions of compact groups, extend this to proper groupoids (thinking of the group action as a groupoid, here). Though this just pushes the question to one about compactness of higher stacks.

Okay, fair enough! I’m glad you agree that it’s a question. (-:

]]>Mike 43: I wouldn’t call them intuitive either. I don’t think your question is specific to this global stuff. Why is it that in equivariant homotopy theory in general, we seem to be only interested in compact (Lie) groups? I don’t have an answer to that either.

]]>@Charles #35: Let’s branch off another thread about ultracommutativity.

]]>@Charles #34: Thanks for those. I need to digest them a little, but none of them are what I would call an *intuitive* reason. They’re all *technical* reasons: some technical mathematical statement goes wrong, or is different, if the groups aren’t compact. But what we’re doing here is just choosing what we want to study. If we had compelling reasons to include noncompact groups, then we would have to include them, and if the mathematics wasn’t quite as pleasant then that would be just too bad. Why are we only *interested* in compact groups?

…but it’s a good idea to remark on what’s standard and what’s not (yet).

]]>I wouldn’t say the nLab is a repository only for well established usage. There’s plenty of material on the cutting edge.

]]>Urs: Yes, by $\mathrm{GlobSp}$ I mean the same thing as $\mathrm{Glob}$, i.e., presheaves on $\mathrm{Orb}$.

David: note that the perspective on “ultracommutative” I offered is a bit of a personal point of view. I’m not sure it would be widely accepted (or widely unaccepted, for that matter).

]]>David C

ultracommutative monoid object

that would be consistent with our naming conventions. Perhaps though, ultracommutative monoid object in a symmetric monoidal (∞,1)-category?

]]>SInce one of the founding ideas of the nLab was to prevent useful exposition being lost in forgotten comments, we should start something on ultracommutativity. But how best to organise it? I see we don’t have a ’commutativity’ page, but rather a lot of ’commutative X’ pages.

So perhaps then ’ultracommutative monoid object’?

]]>