The n-Category Café

Excellent! I’ll suggest to the department that we adopt it as our motto.

We might say that Kato and Manin are projecting outwards (none too carefully) the profound experience of number theorists, expressed very nicely by Minhyong here:

Given the discussion above, one might expect all mathematicians to be number theorists, and maybe in some sense they are. However, there is an even more essential reason why most practicing mathematicians can get by with a rather naive understanding of numbers and might be better off doing so. This is because of the extremely useful geometric picture of the real and complex numbers. Much of the time, it is perfectly reasonable to visualize the complex numbers as a geometric plane, and base all other constructions upon that basic picture, oblivious to the fine structure of our objects, pretty much as one can do plenty of classical physics without worrying about the fact that the macroscopic objects we are considering arise from the complicated interaction of elementary particles. Now, my claim is that the role of a number theorist in mathematics is exactly analogous to the role of a particle theorist in physics. That role being to probe the nature of the ultimate constituents of the objects that others study from a far coarser perspective. Thus, in contrast to the continuum picture of the complex plane, a number theorist is more likely to perceive of each individual number or groups of numbers in a discrete fashion, and in nested hierarchies reflecting various complexities, and even attach a symmetry group to each individual number. It is not that number theorists avoid the plane model, since it is also an important tool in much of number theory. It is just that the plane has a much more grainy and elaborate shape, with many levels of microscopic detail and structure.

As for the physics side of things - Why do we appear to live in a real manifold? - I couldn’t imagine how to proceed.

to which extent stable (∞,1)-categories serve the “same purpose” as (∞,1)-toposes.

Just for completeness, this needs to go along with a nod towards the stable Giraud theorem.

which may provide some insights

I can’t access this right now, for various stupid reasons. One of them being that I go to bed now! :-)

But do you have any particular insight in mind, or is this just meant as a pointer to equivariant bundles in general?

Do you have a pdf of the thing?

Okay, after a very helpful discussion with my wife Megan, who studies equivariant cohomology theory, I think I understand things better. According to the nPOV on cohomology, if $X$ and $A$ are objects in an $(\infty,1)$-topos, the 0th cohomology $H^0(X;A)$ is $\pi_0(Map(X,A))$, while if $A$ is a group object, then $H^1(X;A)= \pi_0(Map(X,B A))$. More generally, if $A$ is $n$ times deloopable, then $H^n(X;A) = \pi_0(Map(X, B^n A)$. In $Top$, this gives you the usual notions if $A$ is a (discrete) group, and in general, $H^1(X;A)$ classifies principal ∞-bundles in whatever $(\infty,1)$-topos.

Now consider the $(\infty,1)$-topos $G Top$ of $G$-equivariant spaces, which can also be described as the $(\infty,1)$-presheaves on the orbit category of $G$. For any other group $\Pi$ there is a notion of a principal $(G,\Pi)$-bundle (where $G$ is the group of equivariance, and $\Pi$ is the structure group of the bundle), and these are classified by maps into a classifying $G$-space $B_G \Pi$. So the principal $(G,\Pi)$-bundles over $X$ can be called $H^0(X;B_G \Pi)$. If we had something of which $B_G \Pi$ was a delooping, we could call the principal $(G,\Pi)$-bundles “$H^1(X;?)$”, but I don’t think there is such a thing. As far as I can tell, $B_G \Pi$ is not connected, in the sense that $∗\to B_G \Pi$ is not an effective epimorphism and thus $B_G \Pi$ is not the quotient of a group object in $G Top$.

Okay, so far so interesting, where do the spectra come in? If we have an object $A$ of our $(\infty,1)$-topos that can be delooped infinitely many times, then we can define $H^n(X;A)$ for any integer $n$ by looking at all the spaces $\Omega^{-n} A = B^n A$. These integer-graded cohomology groups are closely connected to each other, e.g. they often have cup products or Steenrod squares or Poincare duality, so it makes sense to consider them all together as a cohomology theory. We then are motivated to put together all of the objects $\{B^n A\}$ into a spectrum, a single object which encodes all of the cohomology groups of the theory. A general spectrum is a sequence of objects $\{E_n\}$ such that $E_n \simeq \Omega E_{n+1}$; the stronger requirement that $E_{n+1} \simeq B E_n$ restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the $(\infty,1)$-topos. In $Top$, the most “basic” spectra are the Eilenberg-Mac Lane spectra produced from the input of an ordinary abelian group.

Now we can do all of this in $G Top$, and the resulting notion of spectrum is called a naive $G$-spectrum: a sequence of $G$-spaces $\{E_n\}$ with $E_n \simeq \Omega E_{n+1}$. Any naive $G$-spectrum represents a cohomology theory on $G$-spaces. The most “basic” of these are “Eilenberg-Mac Lane $G$-spectra” produced from coefficient systems, i.e. abelian-group-valued presheaves on the orbit category. The cohomology theory represented by such an Eilenberg-Mac Lane $G$-spectrum is called an (integer-graded) Bredon cohomology theory.

Megan and I don’t know whether there is some sense in which Eilenberg-Mac Lane $G$-spectra can be produced by successively “delooping” a coefficient system, but it seems possible.

It turns out, though, that the cohomology theories arising in this way are kind of weird. For instance, when you calculate with them, you see torsion popping up in odd places where you wouldn’t expect it. It would also be nice to have a Poincare duality theorem for $G$-manifolds, but that fails with these theories. The solution people have come up with is to widen the notion of “looping” and “delooping” and thereby the grading: instead of just looking at $\Omega^n = Map(S^n, -)$, we look at $\Omega^V = Map(S^V,-)$, where $V$ is a finite-dimensional representation of $G$ and $S^V$ is its one-point compactification. Now if $A$ is a $G$-space that can be delooped “$V$ times,” we can define $H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A)$. If $A$ can be delooped $V$ times for all representations $V$, then our integer-graded cohomology theory can be expanded to an $RO(G)$-graded cohomology theory, with cohomology groups $H^\alpha(X;A)$ for all formal differences of representations $\alpha = V - W$. The corresponding notion of spectrum is a genuine $G$-spectrum, which consists of spaces $E_V$ for all representations $V$ such that $E_V \simeq \Omega^{W-V} E_W$. A naive Eilenberg-Mac Lane $G$-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a Mackey functor, and in this case we get an $RO(G)$-graded Bredon cohomology theory.

$RO(G)$-graded Bredon cohomology has lots of formal advantages over the integer-graded theory. For instance, the torsion that popped up in odd places before can now be seen as arising by “shifting” of something in the cohomology of a point in an “off-integer dimension,” which was invisible to the integer-graded theory. Also there is a Poincare duality for $G$-manifolds: if $M$ is a $G$-manifold, then we can embed it in a representation $V$ (generally not a trivial one!) and by Thom space arguments, obtain a Poincare duality theorem involving a dimension shift of $\alpha$, where $\alpha$ is generally not an integer (and, apparently, not even uniquely determined by $M$!). Unfortunately, however, $RO(G)$-graded Bredon cohomology is kind of hard to compute.

So I guess what I’m saying now is that the cohomology itself can, and perhaps should, still be viewed using hom-spaces in an $(\infty,1)$-topos. In fact, for nonabelian cohomology, such as the classification of principal bundles with nonabelian structure group, this is the only option. For abelian cohomology that exists at all degrees, we can “package up” the spaces representing it at all degrees into an object called a “spectrum” and then study spectra in their own right. But depending on the $(\infty,1)$-topos we started with, there may be other kinds of “degree” that should be taken into account, in addition to the usual integers.

What’s still not clear to me is whether the notion of “$RO(G)$-grading” is somehow canonically determined by the $(\infty,1)$-topos $G Top$, maybe even in a way that could be applied to other $(\infty,1)$-topoi. Peter May has commented that in some sense, the $RO(G)$-grading isn’t really “fully general” either. There’s a larger class of “homotopy spheres” which are already invertible in the equivariant stable homotopy category, so that all genuine $G$-spectra can be “looped and delooped” by them, and the $RO(G)$-graded cohomology theories they represent could be extended to a grading on that larger class of spheres. But identifying precisely what those spheres are, for any particular $G$, is apparently pretty hard.

I wrote

But this is too restrictive: in the (∞,1)-topos of smooth ∞-groupoids only some of the maps out of $\mathbf{B}G$ have this form.

Jim Stasheff asks

which form?

Jim will know in a moment what I meant, but I say it in full detail anyway, in case anyone else is wondering:

For $G$ a Lie group, regarded as the sheaf on the category $CartSp$ of cartesian smooth manifolds that it represents, the corresponding simplicial sheaf is the usual

$$\mathbf{B}G := \left( \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G\stackrel{\to}{\to} {*} \right) \,.$$

In the standard literature the cohomology of the Lie group $G$ is (equivalently, but usually more implicitly than I say it now) defined simply as the collection of homotopy classes of morphisms of simplicial sheaves $\mathbf{B}G \to A$ from the simplicial sheaf above to some coefficient sheaf.

This is blindly copied from the way one can compute the cohomology of discrete groups: when $G$ is discrete, then $\mathbf{B}G$ happens to be cofibrant in the relevant model category structure, and this is what makes morphisms out of this $\mathbf{B}G$ indeed model cocycles in cohomology.

But for $G$ Lie, the simplicial sheaf $\mathbf{B}G$ above in general fails to be cofibrant. This means that morphisms of simplicial sheaves out of it will fail to see all classes of morphisms in $Ho(\mathbf{B}G,A)$. And this is what cohomology really is.

Notably (for simple, simply connected compact $G$), there is the 3-cocycle in $Ho(\mathbf{B}G,\mathbf{B}^3 U(1))$ whose homotopy fiber is the string 2-group $\mathbf{B}String(G) \to \mathbf{B}G$. This is not a morphism of simplicial sheaves out of $\mathbf{B}G$ directly, one needs a bigger resolution.

For instance let $[\Omega_e G \to P_e G]$ be the evident (up to some non-essential choices) crossed module of based loops and based paths in $G$ and write $\mathbf{B}[\Omega G \to P G]$ for the corresponding simplicial sheaf

$$\mathbf{B}[\Omega G \to P G] := \left( \cdots P G \times P G \times \Omega G \stackrel{\to}{\stackrel{\to}{\to}} P G\stackrel{\to}{\to} {*} \right) \,.$$

Then we have an acyclic fibration $\mathbf{B}[\Omega G \to P G] \stackrel{\simeq}{\to} \mathbf{B}G$, hence this is a somewhat bigger model for $\mathbf{B}G$. Take $\mu_3 \in CE(\mathfrak{g})$ be the canonical Lie algebra 3-cocycle on $\mathfrak{g}$ normalized such that it represents the image in de Rham cohomology of one of the generators of $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$. Then the operation of integrating $\mu$ over 3-balls in $G$ cobounding chosen disks between the loops appearing in $[\Omega G \to P G]$ provides a morphism of simplicial sheaves

$$\array{ \mathbf{B}[\Omega G\to P G] &\stackrel{\int \mu_3}{\to}& \mathbf{B}^3 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G }$$

that represents a class in $Ho(\mathbf{B}G, \mathbf{B}^3 U(1))$ which is not representable by any morphism of simplicial sheaves out of $\mathbf{B}G$ directly.

And the extension of $\mathbf{B}G$ classified by this is $\mathbf{B}String(G)$.

If $G$ had been discrete, the whole construction would have worked by mapping just out of $\mathbf{B}G$ itself. This is what people are used to. This is why one finds sometimes the attitude that cohomology of groups is defined to be morphisms out of the bar-resolution $\mathbf{B}G = G^{\times \bullet}$. But cohomology is more abstractly and more intrinsically defined than that. This is just one algorithm for computing it, applicable in only some situations.

if we are talking about abelian-group-valued $(\infty,1)$-presheaves, then these deloop simply objectwise, no?

Yes, you should be able to do that. What I meant was, I don’t know whether the Eilenberg-Mac Lane G-spectrum associated to a coefficient system is equal to what you would get by regarding the coefficient system as an abelian-group-valued $(\infty,1)$-presheaf and delooping it. Or equivalently, whether the 0-space of the Eilenberg-Mac Lane spectrum associated to a coefficient system is equal to that coefficient system so regarded. It seems possible, but there are so many traps in the equivariant theory that I don’t trust the construction that seems “obvious” to a category theorist to always be the one that the equivariant homotopy theorists are interested in.

From our discussion on the nForum I had gotten away with the impression that Bredon cohomology needs the “non-naive”/”genuine” G-spectra as coefficients instead.

That’s possibly because I myself was under that impression as well. But Megan straightened me out.

it feels as if one should be able to understand this in terms of the “stable Giraud theorem”

What confuses me about the stable Giraud theorem is that it’s claiming that any locally presentable stable $(\infty,1)$-category is a left-exact localization of the stabilization of a category of presheaves on an $(\infty,1)$-category. But the Schwede-Shipley result for stable model categories is about model structures on a category of spectrum-valued presheaves on a spectrally enriched category. In other words, in the latter case the domain category of the presheaves is already “stabilized” in some sense. How are those two related?

Maybe choosing a different adjunction here could account for non-standard grading such as RO(G)-grading?

Something like that may be it. You have the functors $\Sigma^\infty$ and $\Omega^\infty$ switched, by the way; $\Sigma^\infty$ goes to the stabilization and $\Omega^\infty$ from it. I would say that in general the coefficient objects are of the form $\Omega^\infty \Sigma^n A$ for some stable object $A$. (Most $A$ we’re interested in will not be of the form $\Sigma^\infty$ of anything.) We can then get RO(G)-grading by replacing “$n$” by some nontrivial representation sphere in the “genuine” equivariant stable category. Of course then the question is, how do you know that the “genuine” stable category is the adjunction you want to put there.

What confuses me about the stable Giraud theorem is that it’s claiming that any locally presentable stable (∞,1)-category is a left-exact localization of the stabilization of a category of presheaves on an (∞,1)-category. But the Schwede-Shipley result for stable model categories is about model structures on a category of spectrum-valued presheaves on a spectrally enriched category. In other words, in the latter case the domain category of the presheaves is already “stabilized” in some sense. How are those two related?

Yeah, so that’s what I meant: the fact that Schwede-Schipley have

• $Sp$-enriched $Sp$-valued presheaves

instead of

• $sSet$-enriched $Sp$-valued presheaves

must be the fact that accounts for the left-exact localization.

Stable Giraud tells us, tanslated to model category models, that we need a left-Bousfield localization of the model category of $sSet$-enriched $Sp$-valued presheaves. Reading Schwede-Schipley’s result with this in mind must imply that their passage to $Sp$-enrichment (i.e. stabilization of the domain) somehow models this Bousfield localization. In some unfamiliar way. Not that I understand it.

You have the functors $\Sigma^\infty$ and $\Omega^\infty$ switched, by the way;

Ah, right, thanks for catching that.

I would say that in general the coefficient objects are of the form $\Omega \Sigma^n A$ for some stable object $A$. (Most $A$ we’re interested in will not be of the form $\Sigma^\infty$ of anything.)

Ah, right, thanks for cathching that, too!

Of course then the question is, how do you know that the “genuine” stable category is the adjunction you want to put there.

Yeah, it’s still somewhat mysterious. But much less than it was yesterday. I am even feeling inclined to now get back to my orginal motivation of looking at dendroidal presheaves on the orbit category…

http://www.nonduality.com/whatis9.htm
“Brahman is outside time, space, and causality, which are simply forms of empirical experience. No distinction in Brahman or from Brahman is possible.”