## April 4, 2009

### Cohomology

#### Posted by Urs Schreiber

I started writing an entry

This is (supposed to be) a pedagogical motivation of the concepts sheaf, stack, $\infty$-stack and higher topos theory. It assumes only that the reader has a heuristic knowledge of topological spaces and aims to provide from that a heuristic but useful idea of the relevance of the circle of ideas of categories and sheaves, nonabelian cohomology, sheaf cohomology and a bit of higher topos theory.

t should be readable, but will eventually need more polishing. I’d be grateful if readers with very little or no knowledge about sheaves and cohomology could tell me how helpful or not they find this.

In the course of writing this exposition, I also created an entry

about the very general notion in the context of higher $(\infty,1)$-categorical topos theory.

This can also be understood as a contribution to the discussion at Cohomology and Computtion (week 26).

For some details and comments about how the very general discussion relates in particular to familiar abelian sheaf cohomology I created also

And let me recall how these $n$-Lab entries are supposed to work: if you see anything which causes you an urge to improve it, then: hit “edit entry” and do so! But if you make significant changes, please inform us by logging what you did at latest changes.

I also expanded the entry

on my personal $n$Lab web.

Posted at April 4, 2009 6:10 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1944

### Re: Cohomology

Hi Urs,

You say that you are trying to outline the category for Differential Nonabelian Cohomology (“central general abstract nonsense concepts”). By doing this, does this has already made you capable to see anything new regarding M-Theory, that is to discover anything, that no one has ever seen before? If you did, what was it? And if you didn’t, what kind of advantage does that give you?

What it seems to me it is a kind of skelleton of M-Theory. But I cannot see the essence. What is the essence?

Posted by: Daniel de França MTd2 on April 5, 2009 5:26 AM | Permalink | Reply to this

### Re: Cohomology

There’s more to life than $M$-theory.

Posted by: John Baez on April 5, 2009 7:07 AM | Permalink | Reply to this

### Re: Cohomology

“There’s more to life than M-theory.”

Of course there is. And I apologize for making look like so, but since it was late night here, I forgot to what I was trying specificaly to point out. I should had mentioned this dialogue.

Posted by: Daniel de França MTd2 on April 5, 2009 2:06 PM | Permalink | Reply to this

### Re: Cohomology

“There’s more to life than M-theory.”

Not if M-theory is the Theory of Everything. (Sorry, couldn’t resist.)

### Re: Cohomology

I’ll settle for E^2-Theory. The Theory of Everything Else.

Posted by: Jonathan Vos Post on April 6, 2009 7:25 PM | Permalink | Reply to this

### differential cohomology in supergravity

I wrote an entry on the general concept of cohomology, ending with a pointer to aspects of the flavor called differenttial cohomology.

anything new regarding M-theory?

So what’s going on? Let’s try to get this straight:

the full machinery of differential refinements of general cohomology theories, i.e. of general differential cohomology, happens to find an interesting class of applications in what’s called higher gauge theories.

Higher gauge theories are certain functionals, called action functionals, defined on spaces of higher differential cocycles, analogous to and generalizing the way the familiar Yang-Mills theory action functional is a functional on a space of fiber bundles with connection.

A particular class of such higher gauge theories happens to arise packaged in functionals which are called higher dimensional supergravity theories. The full relevance of differential cohomology for these theories was realized only eventually, but then became the very incentive for mathematicians to develop the very theory of generalized differential cohomology in the first place.

Originally the functionals in question were thought of naively as functionals on just higher degree differential forms. A first indication that something more interesting is going on was the understanding of a phenomenon now called the Green-Schwarz mechanism. This essentially said that these functionals have a chance of inducing a well defined quantum theory only if certain differential cohomological obstructions vanished.

Then a bit later it was realized that in closely related contexts more and more subtle differential cohomological effects played a role, related to the fact that some of the higher differential cocycles in question were subject to a peculiar constraint called self-duality. These self-duality constraints are induced by certain quadratic functionals on differential cocycles.

It was the desire to understand precisely these effects in a well-defined mathematical language that made Hopkins and Singer think about differential refinements of general cohomology theories in the first place, and thereby introduce the concept in full generality into mathematics. The seminal article in which this was achieved is

Hopkins, Singer, Quadratic functions in geometry, topology,and M-theory.

This article introduces the general notion of “generalized abelian differential cohomology”, which one may think of as being the theory of abelian (and stable) $\infty$-bundles with connection.

The title alone doesn’t indicate this, though. The title refers to an, apparently important, class of applications of these cohomology theories.

Namely, there is one flavor of the supergravity higher gauge theories, called 11-dimensional supergravity, where the differential cocycles in question are certain 2-gerbes with connection with a funny constraint on them. This 2-gerbe with connection is, for historical reasons, called the “M-theory $C_3$-field”. And it is what gives rise to the title of Hopkins-Singer’s work on generalized differential cohomology theories.

At this point it is important to understand that the supergravity action functional on, among other things, this space of constrained 2-gerbes, was once hoped to be an approximation to a hyothetical physical theory dubbed “M-theory” which was connected with high hopes concerning its physical relevance. Correspondingly deep was the disappointment that followed when these hopes kept waiting to be fulfilled, and the suspicion with which today work is met which purports to concern itsel with “M-theory”.

While this is only reasonable, it remains an unfortunate fact that the term “M-theory” keeps being used synonymously for just plain 11-dimensional supergravity, which is a perfectly sober idea, so sober that pure mathematicians see reason to make it a central topic of some of their articles.

It might be wiser to use more careful terminology. Instead of calling that 2-gerbe with connection the “M-theory $C_3$-field”, it would be better to consistently call it the “11-dimensional supergravity $C_3$-field”.

Whatever we call it, there is an interesting mathematical question to be answered here: what exactly is that funny constraint on this 2-gerbe, which I mentioned, cohomologically?

It’s a certain “quantization condition” which was studied by physicist Diaconescu, Moore and Witten. They tried to understand what that field which looks lik a 3-form locally but is globally something more tricky is exactly. They presented what they called a “model for the $C_3$-field”: a formalization of what exactly it is.

This is tricky, because this 2-gerbe, while itself abelian, is crucially controlled by a nonabelian differential cocycle: a certain principal bundle with connection for a non-abelian structure group.

This suggests that a precise description of the “supergravity $C_3$-field” is not something that abelian differential cohomology can formalize intrinsically. Instead, it suggests that the “model for the supergravity $C_3$-field” should be: a certain cohomology class in nonabelian differential cohomology.

So let me slightly reword Danliel’s question and then anwer is:

Questions: anything new regarding 11-dimensional supergravity?

Answer: yes. It seems we are claiming that we can naturally identify and formalize the supergravity $C_3$-field with its correct quantization condition as a cocycle in nonabelian differential cohomology.

We do this in a slight variation of the way in which we also interpret the original Green-Schwarz mechanism in 10-dimensional supergravity with the existence of a certain twisted nonabelian differential cocycle.

But let me note that, as interested I am in discussing this, it is not really on-topic for this thread here. The dicussion properly belongs to the previous thread Twisted differential String- and Fivebrane structures.

The on-topic discussion here was supposed to be:

for laymen: please let me know how you find what I typed under Heuristic introduction to sheaves, cohomology and higher stacks

(I see from the entry’s history that an anonymous contributor has fixed a bunch of typos. Thanks for that! Very much appreciated.)

for experts: please let me know your opinion, improve and expand on the entries on the general $\infty$-categorical notion of cohmology which I started typing into the $n$Lab, such as cohomology and the sub-entries abelian sheaf cohomology and nonabelian cohomology.

There is also a question hidden there: has anyone formalized the notion of a stable ($\infty,1$)-topos?

Posted by: Urs Schreiber on April 5, 2009 6:51 PM | Permalink | Reply to this

### Re: differential cohomology in supergravity

The ‘heuristic’ entry is looking great. We could do with many more such pages.

I’m not sure how to fix:

the generalization of the notion of space which does accomplish this: the notion of a sheaf in proper interpretation in the general context of space and quantity:

or I’d do it myself.

Posted by: David Corfield on April 5, 2009 8:03 PM | Permalink | Reply to this

### heuristic introduction to sheaves, cohomolohy and higher stacks

I’m not sure how to fix:

[…]

or I’d do it myself.

Ah, right. Thanks for catching that. I had begun going over the text and adding more cross references to other existing entries. I messed up the wording here apparently.

Have now changed this to

there is a very general abstract nonsense way to understand [[sheaf|sheaves]] as generalized spaces in the context of a very general abstract [[duality]] between the notions of [[space and quantity]]. The following is a heuristic way to understand this.

Also added a bit more cross references the other way to the old entry [[space and quantity]].

But I am in a hurry, so this was done a bit hastily again. Please feel free to improve.

Posted by: Urs Schreiber on April 5, 2009 9:40 PM | Permalink | Reply to this

### Re: differential cohomology in supergravity

Hi Urs, good work, it’s very helpful and keep it coming. I liked the introduction to sheaves, got a bit lost in the one section where you apply the sheaf to another sheaf. I added in a paragraph about maps between sheaves, and I inserted a query box.

Posted by: Bruce Bartlett on April 5, 2009 10:07 PM | Permalink | Reply to this

### maps of generalized spaces

got a bit lost in the one section where you apply the sheaf to another sheaf.

Good point, I didn’t properly introduce the notation used at this point, even though it’s meant to be the very intuitive notation:

For $X$ and $Y$ two generalized spaces (realized as sheaves), we simply write $X(Y)$ for the collection of morphisms from $Y$ to $X$!

To make this clear, I have now created a subsection at that entry titled maps between generalized spaces which discusses this in detail.

The point here is to make the notation blur the distinction between “probes” in $X(U)$ and maps $U \to X$. The Yoneda lemma says that these are canonically identified. So we just write $X(Y)$ even if $Y$ is not a test space but a generalized space itself, to denote the collection of “probes” of $X$ by $Y$.

The gain is that this makes the “descent condition for $\infty$-stacks”, which otherwise tends to look like an intimidating concept, become conceptually the easiest thing in the world:

just like an isomorphism $U \stackrel{\simeq}{\to} V$ of ordinary test spaces must induce an isomorphism $Z(V) \stackrel{\simeq}{\to} Z(U)$ of collections of probes of the generalized space $Z$ by probe spaces $U$ and $V$ (this is just functoriality of the presheaf $Z$), so a weak equivalence $X \to Y$ of generalized spaces must induce a weak equivalence $Z(Y) \to Z(X)$ of probes by those. Otherwise the entire “probe” interpretatioon were inconsistent.

Let me know if this helps.

(Unfortunately the formatting of the entry is a bit rough now: I stopped using bulleted lists at some point as the software was choking on somthing I couldn’t identify. But we’ll fix that eventually).

Posted by: Urs Schreiber on April 6, 2009 1:36 PM | Permalink | Reply to this

### Re: differential cohomology in supergravity

The notion of stable $(\infty,1)$-topos is just the notion of accessible stable $(\infty,1)$-category. Remember that an $(\infty,1)$-category $C$ is stable it has finite (homotopy) limits and colimits, a zero object, and if it has the property that a commutative square of $C$ is cartesian if and only if it is cocartesian. First notice that any stable $(\infty,1)$-category is canonically enriched in spectra: this comes from the fact that the universal cocomplete stable $\infty$-category generated by the point (resp. by some $(\infty,1)$-category $A$) is the $(\infty,1)$-category of spectra (resp. of presheaves of spectra on $A$). This implies that the stable analog of $(\infty,1)$-topoi are just the stable left Bousfield localizations of $(\infty,1)$-categories of presheaves of spectra by a small set of maps. Such $(\infty,1)$-categories can be characterized as the accessible stable $(\infty,1)$-categories.

So, as you see, there is no real need to write down the theory of stable $(\infty,1)$-topoi’, because it is essentially the same as the theory of stable $\infty$-categories in general: the stable analog of an elementary $(\infty,1)$-topos’ is just an arbitrary stable $(\infty,1)$-category…

Posted by: Denis-Charles Cisinski on April 6, 2009 11:25 AM | Permalink | Reply to this

### stable (infintiy,1)-topoi

This implies that the stable analog of $(\infty,1)$-topoi are just the stable left Bousfield localizations of $(\infty,1)$-categories of presheaves of spectra by a small set of maps. Such $(\infty,1)$-categories can be characterized as the accessible stable $(\infty,1)$-categories.

So the story is entirely analogous to the way Grothendieck $(\infty,1)$-topoi are precisely the localizations of $(\infty,1)$-categories of presheaves with values in topological spaces.

That’s what i would have thought. But given that one can write a book on these (unstable case), I was wondering if somebody found it worthwhile to consider those (stable case) explicitly.

There is another question in the same vein I have:

it should be interesting to consider (localizations of) $(\infty,1)$-categories of presheaves with values in rational topological spaces.

There should be maps back and forth between Grothendieck $(\infty,1)$-topoi and such rational Grothendieck $(\infty,1)$-topoi, and this should essentially be $\infty$-Lie theory.

Has anyone considered “rational Grothendieck $(\infty,1)$-topoi” explicitly? Or is this also considered too straightforward to deserve explicit mentioning?

Posted by: Urs Schreiber on April 6, 2009 1:57 PM | Permalink | Reply to this

### noncommutative geometry

In his above comment Denis-Charles Cisinki kindly emphasized that

- a “stable $(\infty,1)$-topos” is nothing but an arbitrary stable $(\infty,1)$-category,

- while a “stable Grothendieck $(\infty,1)$-topos” is just an accessible stable $(\infty,1)$-category.

Now I am wondering about the following:

in the Kontsevich-style noncommutative geometry program people try to characterize noncommutative spaces in terms of their derived category of coherent sheaves and then more abstractly in terms of triangulated (dg-)categories etc. Given that all these are models for stable $(\infty,1)$-categories, it seems that one could characterize the Kontsevich-style ncg programe by the slogan:

conceive noncommutative spaces as stable $(\infty,1)$-categories.

In light of the fact we can just as well say then:

conceive noncommutative spaces as stable $(\infty,1)$-toposes.

In this form the slogan would have the advantage that it fits nicely into the (petit-)topos theoretic way of thinking about genralized spaces.

Now I am wondering: a big driving force of much of the Kontsevich-style ncg literature is that the standard examples that one want to model are not exactly sheaves/stacks on the category of formal duals of non-commutative algebras: the sieves which they regard as covering do not satisfy all the axioms of a Grothendieck topology.

Kontesvich and Rosenberg have proposed, therefore, to replace in the context of ncg the notion of sheaves with respect to a grothendieck topology by something more general, called sheaves with respect to a “$Q$-category”. See Zoran Škoda’s entry $n$Lab:Q-categories.

I am wndering how this might fit into the pigger picture of “stable $(\infty,1)$-toposes”. A naive first guess would be simply that this is telling us that non-commutative geometric spaces correspond to non-Grothendieck stable $(\infty,1)$-toposes (i.e. those not arising as localizations of a presheaves of spectra).

Posted by: Urs Schreiber on September 26, 2009 2:58 PM | Permalink | Reply to this

### Re: noncommutative geometry

I don’t know much about all of this stuff, but as I said in the discussion here, I have yet to be convinced that the term “stable $n$-topos” is really justified for any value of $n$. If “stable Grothendieck $(\infty,1)$-toposes” are the same as accessible stable $(\infty,1)$-categories, then why not just say “accessible stable $(\infty,1)$-category”? Or maybe “Grothendieck stable $(\infty,1)$-category”, analogous to “Grothendieck abelian category” in the $n=1$ case.

Also, it seems to me more like arbitrary stable $(\infty,1)$-categories are the “stable” analogue of elementary $(\infty,1)$-pretoposes, in the same way that abelian 1-categories are a stable/abelian version of ordinary pretoposes. But the thing that makes an elementary 1-topos a topos rather than a pretopos or even a $\Pi$-pretopos is its subobject classifier, so it seems to me that an “elementary $(\infty,1)$-topos” should come with some sort of object classifier as well.

Posted by: Mike Shulman on September 26, 2009 6:47 PM | Permalink | PGP Sig | Reply to this

### Re: noncommutative geometry

Thanks, Mike. I want to get a better idea of this myself, so thanks for your input. Here two remarks on what you wrote:

If “stable Grothendieck (∞,1)-toposes” are the same as accessible stable (∞,1)-categories, then why not just say “accessible stable (∞,1)-category”?

So the motivation for my followup above was the observation that the program followed by Kontsevich et al to describe generalized (“noncommutative”) spaces dually by triangulated dg-categories or $A_\infty$-categories (as indicated here) would happen to be entirely in-line with the petit-topos way of thinking about generalized spaces once

- we realize that triangulated dg- and $A_\infty$-cats are nothing but presentations of stable $(\infty,1)$-cats …

- and if it made sense to think of the latter as $(\infty,1)$-toposes.

This arose in discussion with Zoran Škoda on the entry derived algebraic geometry: Zoran emphasized that there are two different schools with two different opinions on what derived geometry is:

the Kontsevich school (in lack of a better name) declares effectively that triangulated dg-cats, $A_\infty$-cats and other models for stable $(\infty,1)$-cats with geometric morphisms between them should be regarded as “noncommutative derived spaces”.

The Lurie school (dito) declares derived spaces to be locally affine structured $(\infty,1)$-toposes, also with geometric morphisms between them (of course).

I was thinking about how both points of view could be understood as being unified somehow. Turning this around, one could take the Kontsevich program and its motivating examples as circumstantial evidence for the reply to your question: “Because stable $(\infty,1)$-categories do behave like generalized spaces much like $(\infty,1)$-toposes do.” Maybe.

Concerning your other remark on object classifiers I’m wondering about that, too. Maybe with a little luck Denis-Charles Cisinski sees us discuss his remark here and comes back to give us a hint.

Posted by: Urs Schreiber on September 28, 2009 8:29 AM | Permalink | Reply to this

### Re: noncommutative geometry

Zoran started collecting some of the definitions of generalized-(noncommutative)-spaces-as-stable-$(\infty,1)$-categories at

Posted by: Urs Schreiber on September 28, 2009 10:46 AM | Permalink | Reply to this

### Re: Cohomology

Is there an account of the conditions for cosheaves as a kind of copresheaf to parallel sheaves as a kind of presheaf? Perhaps this is something for the space and quantity entry or the descent and codescent entry.

Posted by: David Corfield on April 6, 2009 9:40 AM | Permalink | Reply to this

### co-sheaves and codescent

A co-sheaf will be a co-presheaf that satisfies codescent. There is something on codescent at

$n$Lab: descent and codescent

Related blog discussion is at

Codescent and the vanKampen theorem

and

Local nets and co-sheaves

which is meant to identify two applications where (higher) co-sheaves play a role.

In particular there is a conjecture there, which is meant to extend the vanKampen theorem:

The assignment $U \mapsto \Pi(U)$ of fundamental $\infty$-groupoids (for instance page 41 here) to spaces is an $\infty$-co-sheaf (i.e. $\infty$-costack).

Evidence is the higher van Kampen theorem which asserts that $\Pi$ satisfies a strict version of codescent, as well as the result in arXiv:0705.0452 and arXiv:0808.1923 which proves the conjecture for the first and second coskeleton $P_1$ and $P_2$ of $\Pi$ (the path 1-groupoid and path 2-groupoid).

The other blog entry is asking if the AQFT axioms are not maybe to be thought of as having to be enhanced to requiring that the net of observables is really a co-sheaf, not just a co-flabby co-presheaf.

Posted by: Urs Schreiber on April 6, 2009 10:31 AM | Permalink | Reply to this

### Re: co-sheaves and codescent

What do people think of the opinion and reasons expressed here on p. 13:

It is legitimate to wonder if codescent is not merely a form of descent, up to some opposite-category-yoga. We explain now why we consider this as misleading.

Posted by: David Corfield on April 6, 2009 3:29 PM | Permalink | Reply to this

### Re: co-sheaves and codescent

I suppose one way to ask my question precisely is what would a parallel page look like entitled – heuristic introduction to cosheaves, homotopy and higher co-stacks?

You might start with a topological space $S^2$ and tell us about $[S^2, X]$ being the second homotopy group.

Might you then say that a space like $S^2$ is entirely characterized by how one can map it into other spaces?

suppose I dream up a space but don’t tell you which one it is, but I give you hints: for each other space $U$ that you can dream up, I tell you how you can co-probe my space by letting me map my space into your space $U$.

Then there are the consistency checks to tell whether I am telling the truth for all the choices of $U$ and maps between them.

The mapping must be a co-presheaf. And it must also be a co-sheaf, satisfying codescent?

And if all that, why can’t I say

we have every justification then to regard co-sheaves in general as perfectly valid generalized spaces; or more precisely: we regard co-sheaves as rules for how to co-probe generalized spaces, and we take these generalized spaces to be entirely specified by their co-sheaves of co-probes?

In other words, what’s wrong with generalizing spaces by the looking at the way ordinary spaces are co-probed?

You talk about this in space and quantity. I can see the motivation for calling co-presheaves ‘generalized quantities’, but must we view things that way. What’s wrong with thinking about them as a generalization of co-probing by spaces?

Couldn’t I have begun by probing a space with something which might make me call that operation ‘quantitative’, leading me to call presheaves generalized quantities?

There seems to be a conceptual difference between probe and co-probe, resembling the statistician’s sample and classification, but I’m not clear why one is space and the other quantity.

By the way, in space and quantity, presumably in

…we can say that the collection of functions on a generalized space $C$ with values in $U \in S$ is …

that should be $X$ rather than $C$. And, a little lower, in the adjunction between $Spaces_S$ and $Quantities_A$, it should be $Quantities_C$.

Posted by: David Corfield on April 7, 2009 9:14 AM | Permalink | Reply to this

### Re: co-sheaves and codescent

Hi David,

interesting points. I am not sure if there is a first-principle symmetry breaking between sheaves and co-sheaves or if it just always boils down to the fact that one is more used to working from one perspective than from the dual.

For instance, one important point of the in-plot perspective is that it matches so well with the idea that you want some structure (a bundle-like structure) $P \to U$ sitting over your test space, and that you want to think of that as arising from pullback along a classifying map $U \to B something$.

At that “heuristic introduction” entry I kind of try to make the point that it is this perspective which leads one naturally to $\infty$-stacks aka $\infty$-sheaves.

(By the way, I spent the morning polishing and expanding that entry, still not perfect, though, so let me hear your comments).

Now, one might have just as well maybe arrived there from a dual point of view, maybe one is just not used to that.

Notice, however, that usually it is an iteraded back and forth: consider for instance the Moerdijk-Reyes model for synthetic differential geometry:

- then they take generalized synthetic smooth spaces to be sheaves on the category of the co-preshesheaves.

Maybe it’s really a matter of going back and forth between presheaves and co-presheaves sufficiently often such as to obtain a required approximation to Isbell self-duality.

Posted by: Urs Schreiber on April 7, 2009 10:08 AM | Permalink | Reply to this

### Re: co-sheaves and codescent

Urs wrote:

it just always boils down to the fact that one is more used to working from one perspective than from the dual.

of course that depends on how `one’ was brought up, cf. toilet trained

Posted by: jim stasheff on April 7, 2009 3:57 PM | Permalink | Reply to this

### Re: co-sheaves and codescent

Urs wrote

…one important point of the in-plot perspective is that it matches so well with the idea that you want some structure (a bundle-like structure) $P \to U$ sitting over your test space, and that you want to think of that as arising from pullback along a classifying map $U \to Bsomething$.

So were one to write the ‘heuristic introduction to cosheaves, homotopy and higher co-stacks’, would there be anything to take the place of this classifying space?

Is there ever a universal something, $f: X \to Y$, so that for any $U$, (homotopy classes of) maps from $X$ to $U$ classify somethings associated to $U$ by pushing $f$ forward to $U$?

Posted by: David Corfield on April 8, 2009 8:48 AM | Permalink | Reply to this

### Re: co-sheaves and codescent

Ultimately, one can look at $C^op$ just as easily as at $C$, and then sheaves become cosheaves, etc.

However, if one already has developed intuition for $C$ rather than for $C^op$, then there is good reason to use sheaves (on $C$) rather than cosheaves (which are sheaves on $C^op$). The reason is that $C$ embeds in the category of sheaves, whereas $C^op$ embeds in the category of cosheaves. That is, morphisms between representable sheaves go in the same direction as morphisms between their representing objects.

Posted by: Toby Bartels on April 7, 2009 9:58 PM | Permalink | Reply to this

### Re: co-sheaves and codescent

That is, morphisms between representable sheaves go in the same direction as morphisms between their representing objects.

(And in fact, these may be safely conflated, since the Yoneda embedding is fully faithful.)

Posted by: Toby Bartels on April 7, 2009 10:03 PM | Permalink | Reply to this

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