## August 19, 2008

### Spivak on Derived Manifolds

#### Posted by John Baez

Derived categories are an efficient way to “infinitely categorify” a given concept without even needing to know about $\infty$-categories. They don’t do everything! But, they let you go quite far without a vast amount of effort — especially since lot of technology has been developed over the years that you can simply grab off the shelf and use.

For example, the derived category of abelian groups is not a full-fledged approach to dealing with ‘weak abelian $\infty$-groups’: for those you need infinite loop spaces, or in other words, connective spectra. But, there’s a very precise sense in which derived category of abelian groups is a way of handling strict abelian $\infty$-groups and weak maps between these. And that’s pretty good.

So, it’s no surprise that derived categories — and their nonabelian brethren, model categories — are taking over the mathematical universe. And once these get the foot in the door, more general infinitely categorified concepts are sure to follow. For example, we’ve recently seen the rise of derived algebraic geometry, where spectra take the place of abelian groups, and $E_\infty$ ring spectra take the place of commutative rings:

It would be a full-time job just keeping up with this subject.

But now we’re seeing the next phase: derived differential geometry!

I just ran into a thesis on the subject:

People interested in comparative smootheology, convenient categories of smooth spaces, stacks and categorification should take a look! Derived manifolds do a different job than general smooth spaces: they explicitly incorporate homotopical ideas. I doubt either is a substitute for the other. But it should be profitable to compare these ideas.

By the way, Spivak is a student of Peter Teichner, whose work we’ve often discussed here.

Posted at August 19, 2008 6:29 PM UTC

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### Re: Spivak on Derived Manifolds

And a nephew of the more famous mathematical writer Spivak. (Really, I think you should include David’s first name in the post title.)

Posted by: Allen Knutson on August 19, 2008 9:01 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Posted by: John Baez on August 20, 2008 2:00 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Actually, I’m not his nephew, although I used to have on my website a picture of a weird-looking muscle man in a speedo, which I linked to as a picture of “Uncle Mike.”

No relation to Michael Spivak.

Posted by: David Spivak on May 7, 2009 3:30 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

I have to say it. I think he should have called it something like “Calculus on Quasi-Smooth Derived Manifolds.” It was such a perfect opportunity given that is name is Spivak.

Posted by: Matt on August 19, 2008 9:05 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

I feel like rather a wet blanket today:
2. David is his own mathematician, and his writing shouldn’t have to look like an adjunct of his uncle’s.

To try to make up for it I’ll point out this paper:

Unification theorems in algebraic geometry

Authors: E. Daniyarova, A. Myasnikov, V. Remeslennikov

(Submitted on 19 Aug 2008)

Abstract: In this paper, for a given finitely generated algebra (an algebraic structure with arbitrary operations and no predicates) A we study finitely generated limit algebras of A, approaching them via model theory and algebraic geometry. Along the way we lay down foundations of algebraic geometry over arbitrary algebraic structures.

Posted by: Allen Knutson on August 20, 2008 8:55 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

I understand that the omission of David’s first name in the headline apparently touched a sensitive point. So maybe, as an outsider, let me reassure you that when reading the entry this morning I quickly followed the link to the thesis and didn’t have much of a confusion about who the author is.

I think it has been common practice here on the Café to have entries titled “XYZ on ABC” with ABC a topic and XYZ a surname , as befits the need for brevity in a headline.

Posted by: Urs Schreiber on August 20, 2008 11:35 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Allen wrote:

To be honest, I didn’t write the headline in a ‘knowingly misleading’ way. I was just following the standard pattern of our blog headlines, as Urs described.

But now that you’ve got me fired up, I want to write some blog entries with headlines like
McCain on Intergalactic Violence, and Sarkozy on Dope.

Posted by: John Baez on August 20, 2008 9:08 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

I agree with both your principles 1 and 2, Allen. But I don’t think they’ve been contravened.

Urs and John have already addressed principle 1. As for 2, if people see the name Spivak and make a wrong assumption, that’s their own prejudice. As far as I can see, there’s nothing the writers here have done to encourage that assumption. His thesis title is “Quasi-smooth derived manifolds”, so to call the piece “Spivak on derived manifolds” is absolutely straight.

Personally, I assumed that it wasn’t the Spivak famous for his textbooks. I didn’t know if he was still alive, and in any case I thought he’d be too old to be likely to be involved in derived geometry. Which is a prejudice in itself, of course.

Posted by: Tom Leinster on August 21, 2008 12:15 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

The elder Spivak is apparently still alive (born in 1940 according to Wikipedia); he was a student of John Milnor, also still alive. So he’s not too old (for derived geometry or anything else).

When I first met Israel Gelfand (born 1913), he was somewhere in his late 70’s, and going strong in mathematics. I think life in America did him good; when I last saw him (around the year 2000), he seemed even more vigorous than when I first knew him! Mac Lane was another example of someone who kept it going a long time.

Posted by: Todd Trimble on August 21, 2008 9:30 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Right. There’s certainly no upper age limit on doing innovative mathematics.

I knew nothing at all about the elder Spivak except for a vague impression of the era in which his textbooks were written — until yesterday when, like you, I looked him up on Wikipedia. To the extent that I actually thought about it, I’d overestimated his age.

Everyone makes guesses… the crucial thing is whether you remain aware that you might be wrong. Here’s a story. I’m giving a seminar in a mathematics department that I’ve never been to before. My host asks me to go and see the department secretary to sort out my expenses, and points me to the room at the end of the corridor; “Professor X is in there right now, but they won’t mind if you interrupt them”. I go in, and see an oldish man in a colourful sweater talking to a youngish woman in smart dress and heels. Which one’s which? We can all guess, and if you had to bet on it there’d be no doubt as to which guess would be more likely to win. But in reality, it’s wise not to act on that guess, in case you’re wrong.

Posted by: Tom Leinster on August 21, 2008 6:36 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Thanks for the pointer.

Just had a quick look. Seems to be that this is what’s going on:

from the possible starting points of looking at smooth spaces he chooses the out-plots perspective (according to Comparative Smootheology in the tradition of Sikorski, Smith, Mostow) and considers locally ringed spaces.

The infinitization is then imposed on the notion of ring.

As Lawvere taught, concerning “space and quantity”, spaces are presheaves and quantities are co-presheaves. So $\infty$-spaces are presheaves with values in $\infty$-structures, such as presheaves with values in simplicial sets, and $\infty$-quantities are co-presheaves with values in $\infty$-structures.

As David Spivak recalls, $C^\infty$ algebras are defined as multiplicative co-presheaves with values in Sets on Euclidean spaces. $Euclid \to Set$. David Spivak infinitizes this by replacing Sets with simplicial sets and enlarges the domain from Euclidean spaces to manifolds.

The result, copresheaves on manifolds with values in simplicial sets (suitably localized) he calls smooth rings (def 2.1.1 and def 2.1.3).

Then an infinitized smooth manifold, a “derived manifold” is one whose out-plots are smooth rings in this sense, def 4.1.2.

It’s maybe noteworhty that, I think, (co)presheaves with values in simplicial sets is the same as simplicial (co)presheaves, i.e. simplicial objects internal to (co)presheaves.

Similarly for (co)presheaves with values in $\infty$-categories (such as simplicial sets satisfying the Kan condition), which are $\infty$-categories internal to (co)presheaves. That simple change of perspective is conceptually somewhat helpful, since it says for instance that these “smooth rings” are like $\infty$-categories internal to a notion of 1-ring.

Generally I am wondering if the out-plot perspective which is crucial for algebraic geometry is still the preferred point of view for smooth geometry.

Jardine considered homotopy categories of simplicial presheaves (not co-presheaves). Let’s consider these presheaves on manifolds and think of the simplicial sets again as models for $\infty$-categories, then these are $\infty$-categories internal to generalized smooth 1-spaces of the diffeological kind. What would stop me from calling that a “derived manifold”? And how would the two concepts be related?

(And then we can mix both perspectives. Andrew Stacey will certainly want derived Frölicher spaces now…)

Posted by: Urs Schreiber on August 20, 2008 11:28 AM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

There is now an $n$Lab entry on the general theory behind this stuff

Posted by: Urs Schreiber on May 6, 2009 2:50 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

There is also an $n$Lab entry on ordinary

Eventually we’ll get one on those [[$\infty$-$C^\infty$-algebras]].

Posted by: Urs Schreiber on May 12, 2009 6:46 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

I should maybe clarify that the last part of my previous comment was a remark on John’s

Derived manifolds do a different job than general smooth spaces: they explicitly incorporate homotopical ideas. I doubt either is a substitute for the other.

Up to an in-plot/out-plot change of perspective, D.Spivak’s derived manifolds should be like $\infty$-categories internal to general smooth 1-spaces. Smooth $\infty$-categories.

I mean, we had a bit of discussion on smooth $\infty$-categouries in this sense here and elsewhere.

But maybe I am being too optimistic about the equivalence of infinitized in-plot versus out-plot pictures. I am happily working in the in-plot picture, but I understand that derived out-plots are en vogue. If there is a considerable difference, I’d really like to see a comparative discussion of the two.

We need Comparative $\infty$-Smootheology.

Posted by: Urs Schreiber on August 20, 2008 11:49 AM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: What is the right definition of generalized smooth differential graded-commutative algebras?
Tracked: September 9, 2008 11:01 AM

### Re: Spivak on Derived Manifolds

In the course of writing the reference section for [[geometry (for structured $(\infty,1)$-toposes)]] I noticed some changes with David Spivak’s references.

I didn’t track this closely and may be misremembering and/or overlooking some thing, but it seems to me that

• the original pdf version of the thesis is no longer available

• the new version on the arXiv differs from it substantially

• I understand from the remarks in the acknowledgement that the new version fixes some mistakes of the original version, but the new version also seems to miss some more general comments on structured spaces which to me looked well worth keeping, if not in detail, at least in spirit (a remark on this is in the new paragraph 10.1) Is there any stable online reference to the original version? Maybe one including an alert as to which technical aspects were improved by the arXiv version? That would be useful.

• in light of my discussion with Andrew Stacey on (Frolicher) bipresheaves it is noteworthy that the new version is getting quite close to that – now p. 28 defines those bi-presheaves (where for Andrew and my “gros topos” perspective one would take the sites $A = B$ to be equal, both being Euclidean spaces)

• generally one might read some of the modifications as reflecting aspects of the above blog discussion, for instance in section 10.3.

Posted by: Urs Schreiber on July 27, 2009 2:37 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Posted by: Eugene Lerman on July 27, 2009 5:15 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Posted by: Toby Bartels on July 27, 2009 8:00 PM | Permalink | Reply to this

### Re: Spivak on Derived Manifolds

Eugene’s comment source code was missing a closing quotation mark. I fixed it. Now the link should work.

Thanks, Eugene, I now saw where on David’s website the link is hidden.

Posted by: Urs Schreiber on July 27, 2009 8:07 PM | Permalink | Reply to this

### Fukaya category

The Fukaya category of a suitable manifold $X$ has as object Lagrangian submanifolds and its $A_\infty$-hom spaces are built from the intersection of these.

This usually in practice this involves perturbing these Lagrangians such as to make the intersections transverse and is all in all a somewhat messy construction, at least as compared to the simple clean construction of the mirror-dual derived category of coherent sheaves.

In the presented context, one is naturally led to wonder if the theory of derived manifolds would be of help here.

I am imagining that in a suitably “corrected” context we should be able to say that the Fukaya category of $X$ is simply the category whose hom-object between Lagrangians $L_1$ and $L_2$ is an simply the homotopy pullback

$\array{ && L_1 \times_X^h L_2 \\ & \swarrow && \searrow \\ L_1 &&&& L_2 \\ & \searrow && \swarrow \\ && X }$

in the right context.

I can’t claim to have tried to see what this may mean in any technical detail apart from the following plausibility construction:

the 1-cells in $L_1 \times_C^h L_2$ should be paths in $X$ from $L_1$ to $L_2$. But if we somehow demand everything in sight to be holomorphic then all non-constant paths are ruled out, as an odd-dimensional thing cannot be holomorphic. Something like that.

then the 2-cells should be paths of paths whose boundary runs along $L_1$ and $L_2$ and such that everything is holomorphic. Now, this begins to look precisely like the holomorphic disks that indeed define the differential in the hom-space of the Fukaya category.

And so on.

Is there any chance that something like this might contain a grain of truth? Did anyone look at the Fukaya category from the point of view of derived manifolds?

Posted by: Urs Schreiber on July 31, 2009 1:14 PM | Permalink | Reply to this

### Re: Fukaya category

In fact Alan Weinstein and I were talking about this just the other day. There are even much simpler problems involving symplectic geometry where one would naively have a category except for this problem: composition is only well-defined when certain Lagrangian submanifolds or subspaces are in ‘general position’, i.e. transverse to each other. This instantly made me think of David Spivak’s work.

I think it’s good to start with simple examples. Maybe the simplest one is this. One can try to create a category where the objects are finite-dimensional symplectic vector spaces and a morphism

$L : V \to W$

is a Lagrangian subspace of $V^{op} \times W$. (The ‘op’ of a symplectic vector space is the same vector space with its symplectic structure multiplied by $-1$.) We can think of such a morphism as a span

$\array{ && L \\ & \swarrow && \searrow \\ V^{op} &&&& W }$

and I guess naively composition is defined using pullback of spans. But I guess this only works when some subspaces involved are in general position.

I’m a bit mixed up now and too busy to straighten out my thoughts, but this is the rough idea.

Posted by: John Baez on July 31, 2009 7:16 PM | Permalink | Reply to this

### Re: Fukaya category

I guess naively composition is defined using pullback of spans.

My first thought when you say this is to work in a category of generalized smooth spaces where all pullbacks exist. It might be more interesting if sometimes the pullback exists but is not a Lagrangian subspace!

Posted by: Toby Bartels on July 31, 2009 7:30 PM | Permalink | Reply to this

### Re: Fukaya category

Since I was working in the simplified world of linear algebra, and the category of vector spaces has all pullbacks, the problem in that context might be something like this: sometimes the pullback isn’t Lagrangian.

I’d really need to look at a couple of examples to make sure I’m not saying something silly…

Posted by: John Baez on July 31, 2009 7:55 PM | Permalink | Reply to this

### Re: Fukaya category

I think you may be looking for something like this paper:

Functoriality for Lagrangian correspondences in Floer theory

by Katrin Wehrheim and Chris T. Woodward, arXiv:0708.2851

Posted by: Eugene Lerman on July 31, 2009 9:27 PM | Permalink | Reply to this

### Re: Fukaya category

Okay, thanks — that’s the Wehrheim–Woodward paper that Weinstein was telling me about. (These W folks really stick together.)

This paper uses a clever formal trick. Suppose you have a gadget $C$ that’s trying to be a category, but where composition of morphisms is not always well-defined. Then you can make up an honest category $\tilde{C}$ where the morphisms are equivalence classes of formal chains of morphisms in $C$:

$x_1 \stackrel{f_1}{\longrightarrow} x_2 \stackrel{f_2}{\longrightarrow} \cdots \cdots \stackrel{f_n}{\longrightarrow} x_{n+1}$

There’s an obvious formal way to compose these. But then we use an equivalence relation saying that

$x_1 \stackrel{f_1}{\longrightarrow} \cdots \cdots x_i \stackrel{f_i}{\longrightarrow} x_{i+1} \stackrel{f_{i+i}}{\longrightarrow} x_{i+2} \stackrel{f_{i+2}}{\longrightarrow} \cdots \cdots \stackrel{f_n}{\longrightarrow} x_{n+1}$

is equivalent to

$x_1 \stackrel{f_1}{\longrightarrow} \cdots \cdots x_i \stackrel{f_{i+1}f_i}{\longrightarrow} x_{i+2} \stackrel{f_{i+2}}{\longrightarrow} \cdots \cdots \stackrel{f_n}{\longrightarrow} x_{n+1}$

whenever the composite $f_{i+1}f_i$ is defined in $C$.

Sneaky, huh?

They use this trick on page 4 to deal with the fact that the composition of Lagrangian correspondences is only well-defined when a transversality condition holds.

Posted by: John Baez on August 1, 2009 8:45 AM | Permalink | Reply to this

### Re: Fukaya category

My impression is that the Fukaya category is much more subtle to define than somehow accounting for the lack of transversality. The main issue is that it is not local (ie not a stack of A_oo categories), which any reasonable derived correction to intersection theory would be. For example people have been wondering for many years about possible relations between Fukaya categories and categories of modules over deformation quantizations (which also eg contain objects labeled by Lagrangians), but again the latter are local. Tsygan has some interesting ideas about correcting the deformation quantization category to “delocalize” it, which seems like a promising approach, but in any case it’s complicated.

Another thing to say is that for EXACT symplectic manifolds (eg Stein manifolds, like cotangent bundles) one expects a much simpler picture. For cotangent bundles Nadler’s paper gives a beautiful picture of the Fukaya category (of -exact- Lagrangians) as the constructible derived category of the base. Rumor has it Kontsevich has generalized Nadler’s theorem to give a picture for Fukaya categories of all exact (or maybe Stein) symplectic manifolds as constructible sheaves (of modules over some A_oo algebra) on a Lagrangian “core”). That seems to be the best answer to date for an analysis-free description of the Fukaya category.

Posted by: David Ben-Zvi on July 31, 2009 7:55 PM | Permalink | Reply to this

### Re: Fukaya category

My impression is that the Fukaya category is much more subtle to define than somehow accounting for the lack of transversality.

Sure. But when faced with a bunch of problems that all should resolve to one nice answer it could be useful to looking for natural answers to one of them and see if these also help with the rest.

My question/suggestion was actually not so much whether replacing perturbed ordinary manifolds by derived manifolds is a good idea, but whether the homotopy pullback that this suggests might be a way to understand the Hom-complex in the Fukaya category.

Possibly for that to make any sense in the first place one needs to pass beyond derived manifolds to derived stacks, so that these holomorphic homotopies I mentioned have a chance to exist.

Mind you, I haven’t studied the technical detals of the definition of the Fukaya category at all (and am not going to in the foreseeable future) and am just shooting in the dark here. But I thought that question is too obvious not try to bounce it off some experts and see what happens.

Posted by: Urs Schreiber on August 1, 2009 5:08 PM | Permalink | Reply to this

### Re: Fukaya category

IFF one follows the thread back, it becomes clear which is THE Fukaya category under discussion.

Transversality is also crucial in the Chas-Sullivan approach to string topology.
As far as I know, it still doesn’t work for general topological manifolds.

Posted by: jim stasheff on August 1, 2009 12:13 AM | Permalink | Reply to this

### simplicial smooth loci

Did anyone consider (derived) $\infty$-stacks on the simplicial site of cosimplicial Moerdijk-Reyes Loci (i.e. duals of simplicial $C^\infty$-rings)?

Posted by: Urs Schreiber on July 31, 2009 1:26 PM | Permalink | Reply to this

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