The n-Category Café

Thanks. There are so many links to give that it’s hard to fit them all in. (-:

I like that about the Café that a technical discussion can break out in the middle of speculative discussion. In a private e-mail someone observed of the discussion that it “seems to be oscillating between spirituality and higher homotopy theory, which may actually be two names for the same thing.”

But good to have a focused thread here, so let me just ask again, if we really believe in the Baez-Dolan idea that ordinary spectra are a type of $\mathbb{Z}$-groupoid and we now seem to need more exotic gradings by $RO(G)$, does this mean there are $RO(G)$-groupoids?

Is part of the reason for the restriction to $n$-categories that the representations of the trivial group are classified by natural numbers?

So the general data is not only an $(\infty,1)$-topos, but a ringed $(\infty,1)$-topos T […] endowed with the canonical ring $\mathbb{A}^1$ (the affine line).

Yes! I think that’s exactly what I have been saying, as Mike mentioned:

Urs has some suggestions about this here…

I said we want to build the spheres using the given geometric line object (or ring object, if you prefer that).

I like to think of the context of playing all these games with geometric paths and spheres, $\mathbb{A}^1$-homotopies etc. as being any path-structured $(\infty,1)$-topos.

That said, I admit that I follow your comment only up to this point here

The interesting thing is that, when the base scheme is the spectrum of the field of real numbers, the real points functor induces a left Quillen functor from the Nisnevich version of the homotopy theory of schemes to the “genuine” $\mathbb{Z}/2\mathbb{Z}$-equivariant homotopy theory, while the same functor induces a left Quillen functor from the étale verson of the homotopy theory of schemes to the “naive” $\mathbb{Z}/2\mathbb{Z}$-equivariant homotopy theory.

I think I see what you mean, but I am not sure I see how this comes about. Do you have a reference for this? Is this in your motivic articles?

the real points functor induces a left Quillen functor from the Nisnevich version of the homotopy theory of schemes to the “genuine” $\mathbf{Z}/2\mathbf{Z}$-equivariant homotopy theory

Do you really mean the real points functor? I thought it was the complex points that had a $\mathbf{Z}/2\mathbf{Z}$-action coming from complex conjugation.

And $E_\infty$ operads play a role here too: a symmetric monoidal $(\infty,1)$-category can be modelled as an $(\infty,1)$-category equipped with an action by an $E_\infty$ operad. (I’ll assert this as though it’s obvious; I’m unaware that anyone has treated symmetric monoidal $(\infty,1)$-categories this way, though I think Lurie has modelled them using Gamma-spaces, which is pretty close.)

In the updated version of Jacob Lurie’s Commutative Algebra it is done this way. More developments along these lines, such as the stabilization hypothesis in these terms, is in $\mathbb{E}_k$-Algebras.

And if I recall correctly what David Ben-Zvi recounted here once, this update goes back to or was triggered by the PhD thesis of John Francis, who went ahead and defined $k$-fold monoidal $(\infty,1)$-categories as algebras in $(\infty,1)$Cat over $E_k$-operads.

My impression is this event has pushed the announced Spectral Schemes from position 6 further off, with position 6 now being taken by $\mathbb{E}_k$-Algebras . But that’s just me guessing from what I can see around me.

Concerning what $E_\infty$-operads with respect to a given $\infty$-stack $(\infty,1)$-topos are, here is, for the record, my running hypothesis which started this entire discussion with Mike in the first place:

Hypothesis. An $E_\infty$-operad in an $(\infty,1)$-topos of $\infty$-stacks on an $(\infty,1)$-site $C$ is the object in the $(\infty,1)$-category of “dendroidal stacks” on $C$ represented by the objectwise $E_\infty$(Top)-valued dendroidal stack.

More concretely, in model category theoretic terms: let $SSet Cat[C^{op}, dSet]_{proj}$ be the $SSet_{Joyal}$-enriched projective global model category of $SSet$-enriched presheaves with values in dendroidal sets with “the” model structure on dendroidal sets, and let $SSet Cat[C^{op}, dSet]_{proj}^{loc}$ be its left Bousfield localization, using Barwick’s theorem, at the set of all Čech nerve projections for covering families.

Then the $N_d(E_k(Top))$-valued presheaf in there, fibrant-cofibrantly replaced, is the $E_k(Sh_{(\infty,1)}(C))$-operad.

Well, maybe it’s less a hypothesis than a motivating question. But I think it should be true that in a lined $(\infty,1)$-sheaf $(\infty,1)$-topos with line object $I$, the geometric $k$-fold look space objects $[I^k/\partial I^k, X]$ have an action not (just) of an $E_k(Top)$-operad, but of a dendroidal sheaf, along such lines.

This discussion got started when a comment by John Greenlees made me wonder if stable equivariant homotopy theory might be a comparatively simple (hah :-) testing ground for this, due to its comparatively simple underlying site structure.

Reading over this, I see that I didn’t complete my thought.

G-equivariant stable homotopy theory (both “naive” and “non-naive”) are symmetric monoidal, under smash product. This structure is compatable with the unstable symmetric monoidal structure: the suspension spectrum functor is monoidal.

Question: Is equivariant stable homotopy theory “fancy equivariant symmetric monoidal”?

If yes, then equivariant stable homotopy theory should have a norm construction, compatible with the space level norm.

For instance, let G be a finite group. Then if R denotes the line (viewed as a non-equivariant space), then N^G(R), as a G-space, is the real regular G representation V.

Since the spectrum level norm should be compatible, this means that if S^1 is the suspension spectrum of the one point compactification of R, then N^G(S^1) should be the suspension spectrum of the one-point compactification of V; i.e., N^G(S^1) = S^V.

Now, the norm construction is monoidal: on spectra, N^G should take smash products of spectra to smash products of G-spectra. But when we form a stable category, we have to “formally invert” S^1. So what we discover (I think), is that if you want to have a “fancy equivariant symmetric monodial” category of G-spectra, you have to “invert” its norm as well.

In other words, I’d expect that: (i) naive equivariant stable homotopy isn’t fancy equivariant monoidal, (ii) non-naive equivariant stable homotopy is fancy equivariant monoidal. Furthermore, I’d conjecture that non-naive G-equivariant stable homotopy theory is the universal stabilization of unstable G-equivariant homotopy theory which is fancy monoidal.

(This is still a hazy idea for me, I’m entirely sure how to formalize it.)

Charles Rezk explained:

$G$-spaces don’t just form an $(\infty,1)$-category. They form an “$(\infty,1)$-category internal to $G$-spaces”. (A vanilla $(\infty,1)$-category is just a one internal to spaces.)

[…]

How does this work? I’ll give you a presheaf of $(\infinity,1)$-categories on the orbit category of $G$. This assigns to a $G$-orbit $X$ the $(\infty,1)$-category “$F(X) := (G-spaces)/X$”, i.e., the slice $(\infty,1)$-category of $G$-spaces over $X$. […]

Okay, I am following. These presheaves of over-$(\infty,1)$-categories generally seem to be a pretty fundamental thing. When objectwise stabilized, it seems one should/ or may think of them as the “quasicoherent $\infty$-stack of modules” canonically determined by the given context. (Along the lines we recalled at here). I was wondering what I should think of them before stabilization. It seems you are giving one perspective here on that.

But we also have a fancy $G$-equivariant operad in $G$-spaces. Could it be that $F$ is an algebra over the $G$-equivariant operad? […] I’m very sure the answer is yes. To say why would mean talking a lot about the equivariant $E_\infty$ operad, and I’m running of out steam here.

Maybe when you have regained some steam, you can talk about this a bit more? I’d be interested in what you have to say here.

I’d conjecture that non-naive $G$-equivariant stable homotopy theory is the universal stabilization of unstable G-equivariant homotopy theory which is fancy monoidal.

I see. Interesting perspective. I need to think about this. Thanks for sharing this thought.

In fact, $(\infty,1)$-toposes are always internal to themselves.

Is that by any chance a different way of talking about the codomain fibration, a.k.a. the self-indexing, which is the canonical way of talking about a topos $E$ as “a category (= fibration = indexed category) in the world of $E$”?

To say why would mean talking a lot about the equivariant $E_\infty$ operad

Can you at least say what you mean by “the” equivariant $E_\infty$ operad? To use a definite article you apparently have a particular one in mind.

I remarked cryptically:

Is that by any chance a different way of talking about the codomain fibration, a.k.a. the self-indexing

To expand on that a bit for those listening who may not be familiar with it. For an ordinary 1-topos $S$, if you want to do mathematics “in the world of $S$,” then you can do “small mathematics,” working with individual objects like groups and rings, just internal to $S$, but if you want to work with large categories (say, the category of rings in the world of $S$) you have an issue. Of course you can talk about the category of internal ring objects in $S$, but when doing category theory we also need to talk about set-indexed families of objects of categories: for instance when we say that a category is cocomplete we mean that any set-indexed family of objects has a coproduct. But in the world of $S$, the objects of $S$ play the role of “sets,” so we need a notion of “a family of ring objects indexed by an object $X\in S$”. The obvious notion is an internal ring object in $S/X$.

In general, for every naturally defined large category in the world of $S$ there are corresponding categories defined in each slice categories of $S$, and via pullback these categories fit together into a pseudofunctor $S^{op}\to CAT$ (aka an “indexed category”), or equivalently a fibration over $S$. Such a pseudofunctor/fibration $C$ comes with a notion of “$X$-indexed object of $C$” for each $X\in S$, namely the category $C(X)$, or equivalently the fiber of the fibration over $X$. You can then “do category theory” with fibrations/pseudofunctors in the world of $S$ just as you do with ordinary large categories. The fibration representing $S$ itself, i.e. “the category of sets in the world of $S$,” is the codomain fibration, or the pseudofunctor sending $X$ to the slice category $S/X$ (and maps to pullback functors).

Of course, if you want to work with small categories, then you have the other option of just working with internal categories in $S$. In some ways this is “easier,” e.g. you can use the internal logic and talk as if the objects of $S$ were sets. (The 2-internal logic of the category of fibrations aims to make this possible for fibrations as well.) Every internal category represents an indexed category in the same way that every object represents a presheaf, so “small categories give rise to large categories.”

And if you don’t like working with fibrations at all, i.e. with large categories as large categories, you can also do some universe trickery to make them into small ones. For instance, if your topos $S$ is the category of sheaves on some site $D$, then you can consider the topos $S'$ of sheaves on the same site but with values in larger sets (i.e. you assume a universe, so that $S$ be the category of small-set-valued sheaves, and $S'$ the category of large-set-valued sheaves). Then internal categories in $S'$ can be treated as “large categories in the world of $S$.” For instance, I think this is what Street did in his paper “The petit topos of globular sets.” An indexed category, i.e. pseudofunctor $S^{op}\to CAT$ can be turned into an internal category in $S'$ by first restricting it to a pseudofunctor $D^{op}\to CAT$, then “strictifying” it to a strict functor $D^{op}\to CAT$, which is the same as an internal category in the category of functors $D^{op}\to SET$; which in most cases will be objectwise a sheaf, hence an internal category in $S'$.

So when Charles said:

I’ll give you a presheaf of $(\infty,1)$-categories on the orbit category of $G$. This assigns to a $G$-orbit $X$ the $(\infty,1)$-category “$F(X) \coloneqq (G-spaces)/X$”, i.e., the slice $(\infty,1)$-category of $G$-spaces over $X$. Given a map $X \to Y$, the functor $F(X) \to F(Y)$ is the one induced by pulling back along the map.

It sounds very much to me like the above construction: start with the self-indexing $X\mapsto (G-spaces)/X$ of the category $S = G-spaces$, restrict it to the site, i.e. the orbit category $D = \mathcal{O}_G$, then regard that as an internal category in the category $G-SPACES$ one universe higher up.

I want to try to describe the way in which the $(\infty,1)$-category of $G$-spaces is supposed to be monoidal over the “$G$-equivariant $E_\infty$ operad”. I learned about equivariant operads from Rekha Santhanam, and much of my understanding of how this monoidal structure works comes out of conversations with David Gepner.

In what follows, “$G$-spaces” always means the $(\infty,1)$-category (=homotopy theory) of $G$-spaces, in the strong sense; that is, weak equivalences are maps which induces weak equivalences on all fixed point spaces.

Fix a finite group $G$. A “fancy G-equivariant $E_\infty$ operad” $U$ is an operad in the symmetric monoidal category of $G$-spaces (under product), characterized by the property: for each $n$ and each subgroup $K\subseteq G\times \Sigma_n$, $U(n)^{K}$ is either (i) contractible, or (ii) empty, depending on whether (i) $K\cap (\{e\}\times \Sigma_n)$ is the trivial subgroup of $G\times \Sigma_n$ or (ii) isn’t. It’s probably better to think of the case (i) subgroups as being the ones which look like graphs of homomorphisms $H\to \Sigma_n$, where $H\subseteq G$ is a subgroup.

(A side remark: although $U$ is an operad from the point of view of category theory, it isn’t really properly an operad in the $(\infty,1)$-category theory sense. The object $U(n)$ is not merely a “an object of $G$-spaces equipped with an action of $\Sigma_n$”, as you would expect, but rather a more sophisticated thing, “an object of $G\times \Sigma_n$-spaces”.)

I want to use Elmendorf’s theorem freely when talking about $G$-spaces. So if I have a $G$-space $X$, I get a presheaf on the orbit category $O=O_G$ of $G$, by $G/H \mapsto X^H$. Conversely, I can reconstruct $X$ (up to $G$-equivariant weak equivalence) from its associated presheaf.

The notable feature of $U(n)$ is that the $H$-fixed points of the quotient space $U(n)/\Sigma_n$ is the classifying space of the category of functors $H\to \Sigma_n$ (for subgroups $H$ of $G$).

I need to think about a generalization of this. If $X$ is a $G$-space, consider $(U(n)\times X^n)/\Sigma_n$. The $H$-fixed points of this map to the $H$-fixed points of $U(n)/\Sigma_n$, so I get a “bundle”

A point in the “base” represents a functor $\rho\colon H\to \Sigma_n$. The “fiber” over $\rho$ is a space $\mathrm{Hom}_G( S_\rho, X)$ of $G$-equivariant maps. What’s $S_\rho$? That’s a $G$-space which sits in a fibration $S_\rho\to G/H$, with fiber equal to $\{1,\dots,n\}$, and with $G$ action determined by the homomorphism $\rho \colon H\to\Sigma_n$. It’s “determined” in the following way: the group $H$ has to act on the fiber over the coset $eH$, and it acts according to the homomorphism $\rho$.

Some examples: If $\rho(H)=\{e\}$, then $S_\rho = \{1,\dots,n\}\times G/H$, and so $\mathrm{Hom}_G(S_\rho, X)= (X^H)^n$. If $\rho\colon H\to \Sigma_n$ describes the left action of $H$ on itself, then $S_\rho= G$, and so $\mathrm{Hom}_G(S_\rho, X)=X$.

Now let $X$ be an algebra for $U$. The data of such an algebra are maps $(U(n)\times X^n)/\Sigma_n \to X$ of $G$-spaces. I want to understand these in terms of fixed points: $[(U(n)\times X^n)/\Sigma_n]^H \to X^H$.

I’ll continue in another post …

(is the stabilization of a presheaf (∞,1)-category the same as the category of presheaves of spectra?)

Yes, that observation is part of the proof of the stable Giraud theorem, it is recalled as the first of the displayed equations in our section on stable Giraud.

I’d think this follows pretty directly from observing that loop space and suspension objects of presheaves are done objectwise, being pullbacks and pushouts. So also the stabilization works objectwise.

But a genuine $G$-spectrum is not just a particular kind of naive one; it has more data.

Hm, right. But can you help me see this more concretely: how would my functor from left to right (which I just sketched, I am not fully sure here) fail to be full and faithful?

why only study actions of groups? What I mean is a group action is a special kind of a groupoid, an action groupoid. What about a more general groupoids, such as Lie groupoids or their topological analogues (I am not sure what that would mean concretely) or, even more generally, some sort of a geometric stack?

The notion of “naive” $G$-equivartiant spectrum (the obvious definition that comes to mind) has a straightforward generalization to groupoids and beyond.

But historically people at some point decided that naive spectra are too naive, and that something more non-naive is needed. Unfortunately the non-naive version is a bit harder to interpret conceptually. It currently seems to justify its existence mainly due to the fact that it leads to interesting structures.

What we are discussing here is about how one might understand why “non-naive” genuine $G$-spectra are conceptually natural. Charles Rezk proposed that they can be understood as arising from a universal stabilization subject to a certain constraint on a monoidal structure. If that’s the right answer, I am still hoping that we can understand this as an analog of stabilization at geometric “Tate spheres”.

In any case, I gather it is not quite clear yet. When it becomes clear, when we have a good conceptual understanding on “genuine” $G$-spectra I suppose it will be just as straightforward to generalize to groupoid-equivariance as it is for the naive version.

So while I can’t help at the moment with groupoid equivariant spectra, I can point out something about this:

What about a more general groupoids, such as Lie groupoids or their topological analogues (I am not sure what that would mean concretely) or, even more generally, some sort of a geometric stack?

So here is a text that discusses the generalization of genuine $G$-equivariant spectra to the case that $G$ is generalized to an $\infty$-stack of $\infty$-groups modeled on $E_\infty$-rings (how’s that?):

• Clark Barwick, Equivariant derived algebraic geometry and K-theory.

I have maybe not fully absorbed this article yet, but I think I understand what’s going on. It looks to me – but I’d be glad to be corrected – that while this does enormously generalize the classical theory of genuine $G$-equivariant spectra, it does not really shed much light on the conceptual questions we were trying to understand. But maybe I am wrong about that.

By the way, concerning our other discussion about quasicoherent $\infty$-stacks of modules, there is some interesting stuff about this in the last two sections.

Blumberg’s paper looks nice. I’d say what he does is something like this: he identifies equivariant spectra as being functors on (nice) G-spaces which make each G-orbit G/H “look like a dualizable object”. (That’s my interpretation of his Theorem 1.2, anyway.) It’s already known that “inverting representation spheres S^V” makes “G/H dualizable”, so Blumberg seems to have proved the reverse.

That’s really only a new observation for general compact Lie groups G (I think). For finite groups G, equivariant stable homotopy makes G/H look self-dual; this fact about equivariant stable homotopy for finite groups is basic thing that makes the theory of “G-equivariant Gamma-spaces” (as developed by May, Shimakawa, …) work.

Yes, the paper by Costenoble and Waner is the one which talks about “G-operads”, and proves a recognition principle for equivarant loop spaces (for finite G.)

That’s really only a new observation for general compact Lie groups $G$ (I think). For finite groups $G$, equivariant stable homotopy makes $G/H$ look self-dual; this fact about equivariant stable homotopy for finite groups is basic thing that makes the theory of “$G$-equivariant Gamma-spaces” (as developed by May, Shimakawa, …) work.

i see. I have to run now and need to look into this later in more detail.

But can you rephrase this as something like: the stable $G$-equivariant homtopy category is that of the localization of the projective model $Func(G Space_{nice}, sSet)$ at the set of morphisms $xyz$?

I think your rephrasing is correct, where $GSpace_{nice}$ means something like “$G$-spaces which are homeomorphic to finite $G$-CW-complexes” . Blumberg calls this category $W_G$, and for him its an enriched category (enriched over $G$-spaces); his functors are enriched functors, and the target category should be $G$-spaces. (I gather that this can always be rephrased without using $G$-enirched stuff; I think it’s equivalent to use the version of $W_G$ enriched over ordinary spaces (no $G$-action), and ordinary continuous functors to $G$-spaces. But don’t quote me on that …)

I don’t think this claim is very new, though quite possibly it hasn’t appeared in print before. I expect it was already known to experts that if you take functors from $\mathcal{W}_G$ to $G$-spaces, and you localize at a set of maps whose role is to “force representation spheres $S^V$ to be invertible”, then you get the stable homotopy category. (But I’m not an expert, so don’t quote me on that either.)

Mandell and May do something very like this in their paper on orthogonal equivariant spectra, except they use a different category for $GSpace_{nice}$ (a non-full subcategory of $W_G$ constructed from representation spheres).

Blumberg is working with $W_G$, and is proposing a different set of maps to localize with respect to, apparently ones which “make $G/H$ dualizable”.

(I think people have avoided dealing with $W_G$, because they believe that it doesn’t lead to a good model for commutive ring spectra. At least that’s what Mandell, May, Schwede, and Shipley seem to think happens in the non-equivariant case.)

I think your rephrasing is correct,

Thanks, but I am afraid I still don’t understand what I am trying to understand.

One aspect that doesn’t seem to fit what I am hoping to see is that Blumberg and his forerunners us copresheaves/diagrams also with values in $G$-spaces, instead of with values in plain spaces/simplicial sets.

But maybe I should say more explicitly what it is that I am trying to understand.

Namely, I am trying to understand the $G$-equivariant case as the generalization of the following non-equivariant situation (which possibly I also don’t fully understand…):

$$\array{ Top \simeq Sh_{(\infty,1)}(*) &\stackrel{localize at \{\mathbb{R}\times X \to X\}}{\leftarrow} & Sh_{(\infty,1)}(Top) \\ {}^{\mathllap{invert categorical looping}}\downarrow && \downarrow^{\mathrlap{invert geometric looping}} \\ Sp &\stackrel{localize at \{\mathbb{R}\times X \to X\}}{\leftarrow}& ??? }$$

Here on the top right we have $\infty$-stacks on $Top$. Localizing that at $\mathbb{R}$-homtopies yields plain $Top$ (as in Dugger’s notes). Inverting categorical looping in there yields spectra.

But as long as we are still in $Sh_{(\infty,1)}(Top)$ we can and should actually invert geometric looping, in terms of the geometric loop built from $\mathbb{R}$. Because that’s what we know is the right thing to do in other cases like motivic cohomology and, as we seem to have said, should also be the step that explains the “genuine” $G$-spectra.

My trouble starts probably with me not being quite sure how to think of the thing in the bottom right corner obtained this way. But I would expect that localizing it at $\mathbb{R}$-homotopies now turns this into the category of spectra.

Maybe I am wrong about this. With $I$ a line object in an $(\infty,1)$-topos, does localizing at $I$-homotopies commute with inverting $I$-looping?

In any case, my idea would be that we should try to understand the right column of this diagram for the equivariant case, because that seems to be where the geometric looping is natural, whereas after localization it looks like an ad-hoc construction.

So I am looking for a way to make sense o a diagram of the form

$$\array{ G Space \simeq PSh_{(\infty,1)}(O_G^{op}) &\stackrel{localize at ??something??}{\leftarrow} & P Sh_{(\infty,1)}(G Space^{op}) \\ {}^{\mathllap{}}\downarrow && \downarrow^{\mathrlap{invert geometric looping}} \\ G Sp &\stackrel{localize at ??something??}{\leftarrow}& ??? }$$

All diagrams here in big imaginary quotation marks. This is not a claim, but a question.

The thing is that in the $G$-equivariant literature we see the left side of these diagrams discussed. But from $\mathbb{A}^1$-homotopy reasoning etc. we know we want to be working on the right, and we also said that this would explain the non-categorical looping taken on the left, because that should really be the image of the $\mathbb{A}^1$-type geometric looping on the right.

So that’s the picture I am imagining, and of which i am trying to see if or how it is true.

Given what I said above, I figured that if there is an answer along such lines, then it should have shown up in equivariant motivic cohomology.

I probably need to search around a bit more. This here is the first hit on a famous search engine:

• Ben Williams, Equivariant Motivic Cohomology (pdf)

If I see correctly, the definition of equivariant motivic cohomology on p. 5 is Borel equivariant motivic cohomology, namely the cohomology of the action groupoid.

Hm…