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started an entry global equivariant stable homotopy theory with an Idea-section and some references.
I have also created a brief entry for the unstable version: global equivariant homotopy theory.
For anyone who wants to edit and wondering where to add what, let me just highlight that there is the following collection of existing entries (some of them with genuine content, some mostly stubby)
homotopy theory | stable homotopy theory |
---|---|
equivariant homotopy theory | equivariant stable homotopy theory |
global equivariant homotopy theory | global equivariant stable homotopy theory |
In Schwede’s book it says
Besides all compact Lie groups, interesting global families are the classes of all finite groups, or all abelian compact Lie groups.
and
A global family is a non-empty class of compact Lie groups that is closed under isomorphism, closed subgroups and quotient groups.
global equivariant homotopy theory has
What is called global equivariant homotopy theory is a variant of equivariant cohomology in homotopy theory where pointed topological spaces/homotopy types are equipped with G-infinity-actions “for all compact Lie groups G at once”.
Is there any reason for staying at the level of groups? Can’t higher groups feature?
Is there any reason for staying at the level of groups? Can’t higher groups feature?
It’s seems possible (and probably eventually desirable) that you could extend to certain 2-groups, or more.
The word compact seems somewhat crucial here, it turns out. Though you can make some very general definitions for an arbitrary class of groups, a lot of the interesting features of the setup seem to rely in one way or another on having compact groups.
By the way, another reference related to this stuff is the (now old) preprint by Gepner & Henriques, “Homotopy Theory of Orbispaces”.
Thanks, Charles. Would you give us some hints as to your comment on the other thread
global equivariant homotopy theory has cohesion all over the place, it turns out.
Is it a coincidence that Mike Shulman gave global orthogonal spectra as an example of an enriched indexed category along with parameterized spectra, which we believe are cohesive?
Is it a coincidence that Mike Shulman gave global orthogonal spectra as an example of an enriched indexed category along with parameterized spectra, which we believe are cohesive?
I don’t know. The cohesion I am thinking of is an unstable phenomenon; it might be present in a stable situation as well. I can’t figure out the way in which parameterized spectra are supposed to be cohesive, from that discussion.
Coherence for global spaces: there is a theory $\mathrm{Glob}$ of global spaces (one model of which is Schwede’s orthogonal spaces with global weak equivalences). There is an inclusion functor $\Delta\colon \mathrm{Spaces}\to\mathrm{Glob}$ of the usual homotopy theory of spaces (aka $\infty$-groupoids); this inclusion is cohesive.
More generally, for any compact Lie group $G$ there is an object $\mathbb{B}G$ of $\mathrm{Glob}$, and an inclusion functor $\Delta_G\colon G\mathrm{Spaces}\to \mathrm{Glob}/\mathbb{B} G$ into the slice category; this inclusion is also cohesive. Here $G\mathrm{Spaces}$ is the usual $G$-equivariant homotopy theory.
I believe you can generalize this further: for suitable objects $X$ in $\mathrm{Glob}$ which are in some sense “geometric”, it seems you can identify a cohesion in $\mathrm{Glob}/X$. But don’t quote me on that.
Two questions:
Is there an abstrac characterization of the global equivariant (stable/unstable) homotopy category? A universal $\infty$-categorical characterization?
What does the extra left adjoint in the cohesion of the global equivating homotopy category act like? Can you give me an example or else some feeling for what it is?
Re #6
I can’t figure out the way in which parameterized spectra are supposed to be cohesive, from that discussion.
An argument that parameteried spectra as the tangent $\infty$-category to $\infty Grpd$ is cohesive is here.
Urs:
The unstable global equivariant homotopy theory $\mathrm{Glob}$ is easily described: it is presheaves of spaces on $\mathrm{Orb}$, where $\mathrm{Orb}$ is a category (enriched over spaces), which is built from a topologically enriched $(2,1)$-category of compact Lie groups.
Thus, given compact Lie groups $G,H$, let $\mathrm{Rep}(G,H)$ be the groupoid of functors $G\to H$; i.e., objects are continuous homomorphism $G\to H$, and morphisms come from conjugation in $H$. This is in fact a groupoid enriched over spaces, since $H$ has a topology. Now set $\mathrm{Orb}(G,H)=B\mathrm{Rep}(G,H)$, the classifying space of $\mathrm{Rep}(G,H)$. (Alternately, you can regard $\mathrm{Rep}(G,H)$ as a groupoid internal to $\mathrm{Top}$, by using the evident topology on $\mathrm{Hom}(G,H)$, and take the $\mathrm{Top}$-valued nerve of it. It turns out it doesn’t make any essential difference whether you do that or not, a fact which itself relies on $G$ being compact.)
How do we think about all this cohesion? I need some examples of interesting objects of $\mathrm{Glob}$, so let’s note that we have the Yoneda embedding $\mathbb{B}\colon \mathrm{Orb}\to \mathrm{Glob}$, and for each compact Lie group $G$ a fully faithful functor $\Delta_G\colon \mathrm{Top}^G\to \mathrm{Glob}/\mathbb{B} G$, from the usual $G$-equivariant homotopy theory. I write $\Delta_G(X) = \bigl( \delta_G(X) \to \mathbb{B} G\bigr)$ below.
How do the functors $\Pi,\Gamma\colon \mathrm{Glob}\to \mathrm{Top}$ behave on such objects? Well:
$\Pi(\mathbb{B}G) \approx *,\qquad \Gamma(\mathbb{B} G)\approx BG,$and for a $G$-space $X$ (cofibrant, e.g., a $G$ CW-complex), we have
$\Pi(\delta_G(X))\approx X_G,\qquad \Gamma(\delta_G(X))\approx X_{hG},$the honest orbit space and the homotopy orbit space of $X$, respectively.
Thanks, that is useful!
So in particular then by Elmendorf’s theorem the orbit category of a fixed $G$ sits fully faithfully in $Glob/\mathbb{B}G$. I suppose this clears up things for me.
Here is one quick observation, maybe not relevant, but it comes to mind:
there is the $\infty$-topos ETop∞Grpd of $\infty$-sheaves on the category of Cartesian spaces with continuous functions between them. This is cohesive, hence by one of the basic observations of Lawvere, it has besides its canonical enrichment in $\infty$-groupoids another enrichment in $\infty$-groupoids with hom-type being $X,Y \mapsto \Pi[X,Y]$, where $[-,-]$ is the internal hom of the Cartesian monoidal structure and $\Pi$ is the shape modality of the cohesion.
So $Orb$ is the full sub-$\infty$-category of $ETop\infty Grpd$ with the latter equipped with this “cohesive enrichment”, full on the objects of the form $\mathbf{B}G$, for $G$ a compact Lie group.
If I understand correctly what that means, then it is relevant. According to Gepner and Henriques, Glob should be a model of the homotopy theory of orbifolds.
I guess orbifolds are fully faithful inside your infty topos, which I’ll call E. You’ve just explained how to extract a global homotopy type from an object of E. Send X to $\Pi[\mathbf{B}(-), X]$, obtaining a presheaf on Orb.
In other words, if the objects of ETop∞Grpd are “$\infty$-groupoids locally modeled on Cartesian spaces”, then the objects of global equivariant homotopy theory are “$\infty$-groupoids locally modeled on classifying spaces of compact Lie groups”.
@Charles: yes, looking again at their article, what I said in #10 is essentially just that remark 4.3 in Henriques-Gepner is “purely formal” once ambient cohesion has been established;
@Mike: I am not sure if it is locally modeled on classifying spaces, or rather classifying stacks. (?) Also I find myself periodically needing to re-gauge how I think of the fact we we have just $\infty$-pre-sheaves (albeit on a genuine $\infty$-site) instead of $\infty$-sheaves over $Orb$, and what it means that the mapping spaces of Orb are taken to be $\Pi[X,Y]$ instead of $\mathbf{H}(X,Y)$. Moreover, by Henriques-Gepner this is equivalent to just topological stacks…
Oh, yes, I should have said something like that. The role of the topology on a compact Lie group is indeed confusing; sometimes it seems to be honest topology and sometimes it seems to be homotopical.
There is a useful topology on $\mathrm{Orb}$: coverings are families of homomorphisms $\{G_i\to G\}$ such that at least one the elements of the family is a surjective homomorphism.
The objects such as $\mathbb{B} G$ and $\delta_G(X)$ that I talked about earlier all turn out to be $\infty$-sheaves with respect to this topology. So it is possible that the real gadget of interest here is actually this sheaf $\infty$-category.
This is in fact a groupoid enriched over spaces, since $H$ has a topology. Now set $\mathrm{Orb}(G,H)=B\mathrm{Rep}(G,H)$, the classifying space of $\mathrm{Rep}(G,H)$. (Alternately, you can regard $\mathrm{Rep}(G,H)$ as a groupoid internal to $\mathrm{Top}$, by using the evident topology on $\mathrm{Hom}(G,H)$, and take the $\mathrm{Top}$-valued nerve of it. It turns out it doesn’t make any essential difference whether you do that or not, a fact which itself relies on $G$ being compact.)
When I read something like this, my immediate reaction is “most likely one of these two is the correct thing to do, and the other one only happens to agree with it by accident in the compact case”. If you try to do this with noncompact groups, then I’m sure other things will also go wrong, but is it at least clear which (or neither) of these two definitions is “correct”?
So what does the discovery of this global equivariant cohesiveness tell us about the space of cohesion?
Where Schwede writes
Looking at orthogonal spectra through the eyes of global equivalences is a bit like using a prism: the latter breaks up white light into a spectrum of colors, and global equivalences split a traditional, non-equivariant homotopy type into many different global homotopy types,
could one say something similar for any form of cohesion?
Is there a ’super’ version of ETop∞Grpd, from which there could then be a global equivariant homotopy in terms of compact super Lie groups?
When I read something like this, my immediate reaction is “most likely one of these two is the correct thing to do, and the other one only happens to agree with it by accident in the compact case”.
My guess is that neither is exactly “right” for non-compact groups. But really, the problem is that it’s not clear what “right” means here.
Let’s go through a test case: $\mathrm{Rep}(\mathbb{R},G)$, where $G$ is a Lie group. This is a groupoid. The objects correspond (by the exponential map) to elements $X\in \mathfrak{g}$ of the Lie algebra of $G$; there is a morphism $X\to Y$ for every $g\in G$ such that $\mathrm{ad}(g)(X)=Y$. That is, it is the action groupoid of the adjoint action of $G$ on $\mathfrak{g}$. That’s certainly a good thing.
If I use the discrete topology for objects (but the induced topology for morphisms), then I get a groupoid $\mathrm{Rep}^d(\mathbb{R},G)$ whose isomorphism classes are the adjoint orbits (of which there are typically uncountably many), and whose automorphism groups are the stabilizer subgroups (as topological groups).
If I topologize the objects to obtain $\mathrm{Rep}^t(\mathbb{R},G)$, it is equal to the topological groupoid $\mathfrak{g}\times G \rightrightarrows \mathfrak{g}$ associated to the adjoint action.
The second choice seems to keep more information, so perhaps we should say that 2. is the right choice.
But remember that what I actually did was to take the classifying space of the groupoid, and what I care about is the homotopy type. For choice 1., this will give
$B\mathrm{Rep}^d(\mathbb{R},G) \approx \coprod_{[X]} BG_X,$a disjoint union over representatives of adjoint orbits, of classifying spaces of stabilizer groups.
For choice 2., I discover that
$B\mathrm{Rep}^d(\mathbb{R},G) \approx BG;$there is a one parameter family $F_t$ of endofunctors of $\mathrm{Rep}^d(\mathbb{R},G)$, given on objects by $F_t(X)=tX$, which connects the identity functor with the projection functor $\mathrm{Rep}^d(\mathbb{R}, G)\to G$.
From this point of view, the Lie group $\mathbb{R}$ is indistinguishable from the trivial group, since $B\mathrm{Rep}(\{e\}, G)\approx B G$ as well.
Which of these looks more right?
@Charles, re #15, concerning sheaf conditions: ah, I see, that’s interesting.
One thought: if we think of Orb as a full sub-category of topological $\infty$-stacks on the deloopings $\mathbf{B}G$, as we discussed, then one is tempted to induce a kind of “higher canonical topology”. The standard “canonical topology” which has effective epis as coverings is not interesting here, since $\mathbf{B}G$ is connected, but maybe one could ask coverings to be 1-connected maps. I’d need to think more about it, but that should be related to the coverings that you mention.
@David C, re #17 concerning the space of cohesion, two comments:
first, my remark in #10 was to say: the definition of $Orb$ and hence of of the global equivariant $\infty$-topos $PSh_\infty(Orb)$ has a direct analog in any ambient cohesive $\infty$-topos different from E-topological $\infty$-groupoids (we can also say “compact space”, abstractly). Maybe Charles’s argument for $PSh_\infty(Orb)$ being cohesive goes through in all these cases? (I haven’t seriously thought about it at all yet.)
second, more generally on further models of cohesion: we should probably explore more the possibility of having cohesion over different base $\infty$-toposes. I suppose under mild conditions on an $\infty$-topos $\mathcal{B}$ we can for instance form $Sh_\infty(SmoothMfd, \mathcal{B})$ (hence $\infty$-sheaved but with values in $\mathcal{B}$) and it should again be cohesive, now over $\mathcal{B}$. (Again, I haven’t seriously thought about this yet.)
It’s been pointed out to me that this seems to be what is in fact already going on in
(looking notably at section 4 there).
@Charles #19: Interesting! So why did you take the classifying space of that groupoid? (-: Your option 2 is, I believe, the internal-hom in $ETop\infty Gpd$ between deloopings of Lie groups, which is a perfectly respectable object.
@Urs #21, can we take it as a large hint that whenever there’s something cohomological happening, and then differential refinement takes place, then cohesion is in the air?
Cohesion will be represented in some indexed form as $(\infty, 1)$-toposes cohesive over a base, for which there may be some general constructions of stabilisation, subtoposes of compact objects, …?
By the way, since you’re in a super-frame of mind this week, is there a ’super’ version of ETop∞Grpd?
Mike (22):
The point is to extract some kind of homotopy type from an object of $E=\mathrm{ETop}\infty\mathrm{Gpd}$.
Start with a class of Lie groups.
Consider $\mathrm{Orb}$ as I defined it: objects are groups in my class, morphisms are $\Pi[ \mathbb{B}G, \mathbb{B}H]$ as Urs explained. This $\mathrm{Orb}$ is enriched over spaces, and we get a functor $E\to \mathrm{Func}(\mathrm{Orb}^{op}, \mathrm{Top})$ by sending $X\mapsto \Pi[\mathbb{B}(-),X]$.
Now consider $\mathrm{Orb}'$, with the same objects, but morphisms are $[\mathbb{B}G, \mathbb{B}H]$; this one is enriched over $E$. We get a functor $E\to \mathrm{Func}_{E}(\mathrm{Orb}'^{op}, E)$ by sending $X\mapsto [\mathbb{B}(-),X]$.
Let me now choose a conveniently small class of Lie groups: the one consisting only of the trivial group. Then case 1 amounts to giving the functor $\Pi\colon E\to \Top$, while case 2 amounts to giving the identity functor $E\to E$.
So in case 1, I have extracted some kind of “underlying homotopy type” from an object of $E$, while in case 2 I have accomplished nothing.
Urs. Perhaps you are asking the question: does the cohesion $\mathrm{Top}\to E$ factor through a pair of cohesions
$\mathrm{Top} \to \Func(\mathrm{Orb}^{op}, \mathrm{Top}) \to E.$I’ve shown the first inclusion is cohesive, and you’ve shown that the composite is cohesive. Is the second inclusion cohesive? (On reflection: how would you define the second inclusion?)
I think of the distinctions that show up here as analogous to the difference between etale and Nisnevich topologies that are important in motivic homotopy theory and algebraic $K$-theory. Famously, algebraic $K$-theory does not have etale descent, but it does have Nisnevich descent (under some conditions which I’ve forgotten).
I’m thinking of $\Pi_{et}\colon E\to \mathrm{Top}$ (left adjoint to inclusion) as capturing “etale homotopy type”, while $\Pi_{Nis}\colon E\to \Func(\mathrm{Orb}^{op}, \mathrm{Top})$ (the functor described in previous posts) captures something like “Nisnevich homotopy type”. I’m interested in equivariant cohomology theories like equivariant $K$-theory, which do not have etale descent (i.e., do not factor through $\Pi_{et}$), but do appear to have “Nisnevich descent” (i.e., do factor through $\Pi_{Nis}$).
@Charles: When you describe it that way, then as Urs said, it’s hard to see why you’re using $\Pi$ rather than $\Gamma$. That is, $E$ (like any $(\infty,1)$-category) has a Yoneda embedding, and so there is a straightforward functor $E \to Func({Orb''}^{op},Top)$ where $Orb''$ is the full sub-$(\infty,1)$-category of $E$ on the objects $\mathbb{B}G$. Why apply $\Pi$ to the hom-objects?
Um, because that’s not what I want? Gamma gives simmering more like the underlying point set of an object, rather than the underlying homotopy type.
I have trouble thinking of $\Pi$ as something “underlying” in any sense – it seems more destructive than forgetful, collapsing the topology and the homotopy into one. But I guess that’s just language. Probably I would understand better if I understood what the end result is for. (-:
The main motivations (or at least, my main motivations), are (1) to be able say useful things about globally equivariant homology and cohomology theories, and (2) because global homotopy theory seems to be a good context for dealing with ultracommutativity.
Is there an intuitive reason, from either of those perspectives, why compactness of the groups should be important?
After some consideration, I’m inclined to think that your suggestion (2) in #19 is the more categorically sensible choice. It amounts to considering $Orb$ as a full $E$-enriched subcategory of $E$, taking the restricted Yoneda embedding $E \to [Orb^{op},E]$ of $E$-enriched categories, then applying the strong monoidal functor $\Pi$ (which I prefer to call $ʃ$) to get down to ordinary ($\infty$-groupoid-enriched) $(\infty,1)$-categories and profunctors. If $\mathbb{R}$ is indistisguishable from a point, is that so bad? If what you want is to extract a homotopy type in some $ʃ$-like way, then I don’t find that too surprising. After all, it’s certainly true that $ʃ \mathbb{R} = 1$ as a space, so why not also as a group?
Another random thought which occurs to me: what if instead of $E$-enriched categories we did something similar with $E$-indexed categories? Often in topos theory those are the right thing to look at.
Charles,
sorry for kind of dropping out of this conversation. I am still very interested, but had too much other stuff distracting me.
Let me return a favor: if I have the occasion that people want me to list examples of cohesive $\infty$-toposes (as is happening this week in Münster) I might enjoy to briefly mention/announce that you found one more interesting model. What would be a good way to refer to this? Should I just say “Charles Rezk has something in preparation” or can I announce more, such as a title?
global homotopy theory seems to be a good context for dealing with ultracommutativity
What is the main idea behind ultracommutativity?
Urs: I hope to post something soon (which for a while, will just be a set of notes on my homepage or something, probably titled “Notes on global homotopy theory” or something like that).
Mike: I’ll note that the same argument that makes $\mathbb{R}\approx 1$ works for any nilpotent Lie group.
Why is compactness important? I can’t see a single reason. Here are some places where it crops up.
Sometimes you need to know that a homomorphism $H\to G$ factors as a surjective homomorphism followed by an inclusion of a closed subgroup; compactness ensures this. This is crucial in the construction of the functors forming the cohesion $G\mathrm{Top}\to \mathrm{GlobSp}/\mathbb{B}G$; the above factorization lets you build a left adjoint to the inclusion $\mathcal{O}_G\to \mathrm{Orb}/G$ of the classical orbit category of $G$ into the slice theory of the global orbit category over $G$.
Nearby homomorphisms from compact Lie groups are conjugate. This means that if $\Rep(K,G)$ is the topological groupoid of homorphisms $K\to G$, the classifying space $B\mathrm{Rep}(K,G)$ has the homotopy type of a disjoint union $\coprod_{[\alpha]} BC_G(\alpha)$ over conjugacy classes of homomorphisms $\alpha\colon K\to G$. This is perhaps not strictly necessary for the theory, but it is a striking fact, which is quite different for what happens for certain non-compact Lie groups, e.g., nilpotent ones, which tend to have a “continuous spectrum” rather than a discrete one. (Though perhaps this is still true if $K$ is semi-simple but not compact. I don’t really know.)
If $K$ is compact, and $G\to H$ is a surjective homomorphism of Lie groups, then $\Rep(K,G)\to \Rep(K,H)$ is a fibration of categories. You need this to show that $\Pi_G\colon \mathrm{GlobSp}/\mathbb{B}G \to G\mathrm{Top}$ is product preserving. (For this one, it is likely enough for $K$ to be semi-simple.)
Stable versions of this theory need some kind of “universe”, which is a chosen family of representations of the groups in question. In practice, the universe has an inner product, which limits you to orthogonal representations, and thus effectively to compact groups. It would be good to investigate closely to what extent such an inner product is really needed.
David:
Note: I’m stealing the word “ultracommutative” from Stefan Schwede (who uses it for his “ultracommutative ring spectra”), but I think this is not unjustified: it is a special case of a broader idea.
“Ultracommutative” means exactly “more commutative than commutative”. There is a perfectly sensible notion of commutativity in $(\infty,1)$-theory, which indeed deserves to be called “commutative”, but it is apparently not the final word.
To be precise, think about “commutative monoid objects”. A commutative monoid object $M$ is an object in a category $C$ equipped with maps $1\to M$ and $M\otimes M\to M$, satisfying the expected axioms. Evidently, to be able to make such a definition, $C$ itself must be equipped with a symmetric monoidal structure.
In the $(\infty,1)$-categorical paradise we live in today, we can talk about “symmetric monoidal $(\infty,1)$-categories”, and hence “commutative monoid objects” within them. When you examine this in classical terms, you typically discover an $E_\infty$-operad lurking. For instance, spaces are a symmetric monoidal $(\infty,1)$-category (corresponding to cartesian product). The $(\infty,1)$-categorical commutative monoids then turn out to be the “$E_\infty$-spaces”. More generally, $(\infty,1)$-categorical commutative monoids in a decent Quillen model category usually turn out to be $E_\infty$-algebras for an $E_\infty$-operad.
You can take this up a level: a symmetric monoidal $(\infty,1)$-category may itself be modelled as: an $(\infty,1)$-category (modelled as a quasicategory or complete Segal space, say), with an action by an $E_\infty$-operad.
Okay, that’s commutativity in the $(\infty,1)$-world. It is sufficient for a great many purposes, but there are a few important “classical” examples that don’t fit:
In the ur-$(\infty,1)$-category of topological spaces, we can consider actual commutative monoids, rather than $E_\infty$-ones. We obtain the homotopy theory of topological commutative monoids, which of course are of great significance in algebraic topology (since ordinary homology is basically the free abelian group on a space).
Note that any topological commutative monoid is a fortiori an $E_\infty$-space. Thus, topological commutative monoids are “more commutative” than commutative-in-the-$(\infty,1)$-sense.
In equivariant topology, there is (for finite groups $G$), a notion of a $G$-equivariant $E_\infty$-operad; I’ll call them $E_\infty^G$-operads. Despite appearances, this is a new concept; an $E_\infty^G$-algebra is not merely an $E_\infty$-algebra object in $G$-spaces. Equivariant infinite loop spaces are naturally $E_\infty^G$-algebras; as, in a different way, are the main models of equivariant commutative ring spectra.
It is true that an $E_\infty^G$-algebra is a fortiori an $E_\infty$-algebra. Thus, $E_\infty^G$-algebras are also “more commutative” than your usual run-of-the-mill-$(\infty,1)$-categorical-commutative.
It appears that you can take this idea “up a level”. For instance, the homotopy theory (aka, $(\infty,1)$-category) of $G$-spaces is certainly symmetric monoidal: it is an $E_\infty$-algebra in $(\infty,1)$-categories. However, it is not merely so: it is also an $E_\infty^G$-algebra. (At least, it appears to be … I ought to nail down a proof someday. Or has someone done this already?) This additional $E_\infty^G$-algebra structure encodes additional constructions, which are usually called “norm” functors. In the stable equivariant context, these are the norm functors Hill, Hopkins, and Ravenel used in their solution of the Kervaire invariant problem.
Anyway, all these are examples of “ultracommutativity”. Global homotopy theory gives more. It appears that ultracommutativity comes in lots of different flavors; I don’t have a decent taxonomy of them.
You can read some of my older blatherings on this topic in the comments to this old n-category cafe post on equivariant stable homotopy theory.
Remind me what $GlobSp$ is?
If I may: $GlobSp$ must be what in #9 was $Glob$, the $\infty$-presheaves on the global orbit 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’?
David C
ultracommutative monoid object
that would be consistent with our naming conventions. Perhaps though, ultracommutative monoid object in a symmetric monoidal (∞,1)-category?
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).
I wouldn’t say the nLab is a repository only for well established usage. There’s plenty of material on the cutting edge.
…but it’s a good idea to remark on what’s standard and what’s not (yet).
@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?
@Charles #35: Let’s branch off another thread about ultracommutativity.
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.
Okay, fair enough! I’m glad you agree that it’s a question. (-:
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.
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
$E_{\mathrm{Glob}} = \mathrm{Sh}(\mathrm{LieGpd}),$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.
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 put a working document discussing cohesion in global homotopy theory on my homepage.
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…
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?
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?
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.
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.
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:
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
$\mathrm{Map}_{\mathrm{Top}_{\mathrm{Orb}}}( \delta_H(X), \mathbf{B}G) \approx \mathcal{P}_G(H\curvearrowright X),$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).
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 $\mathrm{OTop}\infty\mathrm{Grpd}$.
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.)
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.
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.
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