February 2, 2010

Derived Synthetic Differential Geometry

Posted by Urs Schreiber

Ordinary synthetic differential geometry – at least the well adapted models – is concerned with 0-truncated generalized spaces that are modeled on smooth loci: the formal duals of finitely generated $C^\infty$-rings.

Under derived synthetic differential geometry I suppose we should want to understand the study of the notions of space that are induced from the geometry (in the sense of geometry for structured $(\infty,1)$-toposes) that is the geometric envelope of the pregeometry constituted by $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, with one of its familiar site structures.

This would seem to be an excellent candidate for the ambient geometry in which most of fundamental physics, as presently conceived, takes place.

Maybe we can chat a bit about it here.

For convenience, I repeat the above paragraph, with hyperlinks to background information included:

Ordinary synthetic differential geometry – at least the well adapted models – is concerned with 0-truncated generalized spaces that are modeled on smooth loci: the formal duals of finitely generated $C^\infty$-rings. Under derived synthetic differential geometry I suppose we should want to understand the study of the notions of space that are induced from the geometry (in the sense of geometry for structured $(\infty,1)$-toposes) that is the geometric envelope of the pregeometry constituted by $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, with one of its familiar site structures.

the approach sketched in the very last paragraph of Structured Spaces and then carried out in some detail in

goes in the direction of derived synthetic differential geometry . But it is different from what is indicated above:

• following Lurie, David Spivak considers, essentially , the pregeometry given by $\mathcal{T} :=$ CartSp. This is the formal dual of free finitely generated $C^\infty$-rings.

• the above suggests to use $\mathcal{T} := (C^\infty Ring^{fin})^{op}$, the formal dual of all finitely generated $C^\infty$-rings.

I understand well how the choice of pregeometry in the above article is motivated, and have no quarrals with that, I also understand that $CartSp$ is not in fact quite a pregeometry, as discussed in the article, and I understand that in the original version of the thesis instead Diff is used, which is a pregeometry and is conjectured to yield an equivalent result as that discussed in the arXiv version. Still, $Diff$ is just a full subcategory $Diff \hookrightarrow (C^\infty Ring^{fin})^{op}$. I just think for the record this is a point that deserves some highlighting, obvious as it may be.

In fact, it would seem that we could go through sections 4.2 and 4.3 of Structured Spaces, and pretty much systematically replace the sites of ordinary rings appearing there by the corresponding sites of $C^\infty$-rings. Can’t we?

Posted at February 2, 2010 11:42 PM UTC

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Re: Derived ???

Is this derived’ in a new sense or are notions of up to homotopy’ or resolution by a suitably free object hiding here somewhere?

Posted by: jim stasheff on February 3, 2010 1:00 PM | Permalink | Reply to this

Re: Derived ???

Is this ‘derived’ in a new sense or are notions of ‘up to homotopy’ or resolution by a suitably free object hiding here somewhere?

Thanks for the question, Jim.

I used to complain, here and elsewhere, that the term “derived xyz-geometry” is unfortunate in its undescriptiveness (and not because of the xyz :-). But it has become so well established and popular that I figured if I want to make myself as well understood as possible in the space that I have for a blog entry headline, then I’d better do it that way.

I started writing a reply to your question in the entry higher geometry in the section derived geometry.

You’ll meet old friends there, but now at their home. At their natural home, as one says. ;-)

Posted by: Urs Schreiber on February 3, 2010 2:57 PM | Permalink | Reply to this

Re: Derived ???

Yes, that helps - it’s the limits of (or unimaginative use of) language that gave me pains. So derived geometry
means a geometric interpretation/analog of derived in the sense of homological resolutions.

Posted by: jim stasheff on February 4, 2010 1:31 PM | Permalink | Reply to this

Re: Derived ???

derived geometry means a geometric interpretation/analog of derived in the sense of homological resolutions.

Yes, that’s certainly one of the main motivating points: we don’t just want to algebraically form homological resolutions, but we want to interpret the resolutions themselves as dually having a geometric incarnation. This geometry is “derived geometry”.

Posted by: Urs Schreiber on February 4, 2010 2:08 PM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

Could you give us a sense of the difference it might make for synthetic differential geometry working with all finitely generated $C^\infty$-rings rather than just the free ones?

You mention at derived smooth manifold that Spivak is using something related to the Isbell envelope. Todd was indicating to us here that he knew of “a general biclosed bicategory niche into which things like MacNeille completions and Isbell envelopes fit”. I wonder if his niche is relevant to Spivak’s geometry.

Posted by: David Corfield on February 4, 2010 10:26 AM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

Could you give us a sense of the difference it might make for synthetic differential geometry working with all finitely generated $C^\infty$-rings rather than just the free ones?

A free $C^\infty$-ring is one of the form $C^\infty(\mathbb{R}^n)$ – smooth functions on a cartesian space.

So ordinary sheaves on the site of formal duals of free $C^\infty$-rings are sheaves on the category of manifolds. These contain for instance diffeological spaces (as the full subcategory of concrete sheaves), but they do not contain the crucial ingredient of a smooth topos: a nontrivial infinitesimal interval $D = \{x \in R| x^2 = 0\}$.

Similarly, $\infty$-sheaves ($\infty$-stacks) on the category of formal duals of free $C^\infty$-rings are $\infty$-sheaves on the category of manifolds. This contains all $\infty$-Lie groupoids, but not for instance $\infty$-Lie algebroids - which are $\infty$-Lie groupoids with “infinitesimal morphisms”.

These infinitesimal objects enter the picture when we test not just on formal duals of free $C^\infty$-rings, but on all of them. Actually, we don’t need to test on all of them, it is sufficient to test on those that are the product of a free ring with a “Weil algebra”: a smooth ring of functions on an infinitesimal neighbourhood of a point.

Similarly when we pass to derived stacks here, by generalizing our test objects to cosimplicial test objects, i.e. our rings to simplicial rings: “$\infty$-$C^\infty$-rings”, if you wish.

For instance we may want to regard the BV-BRST complex as the Chevalley-Eilenberg algebra of a derived $\infty$-Lie algebroid. As such it exists in $\infty$-stacks on formal duals of simplicial $C^\infty$-rings, but not – I think – in $\infty$-stacks on just formal duals of simplicial free $C^\infty$-rings. Again, it is the notion of infinitesimal that is otherwise missing.

So the geometry $\mathcal{G}$ (in the sense of geometry for structured $(\infty,1)$-toposes) for a derived version of synthetic differential geometry should be the formal duals of simplicial $C^\infty$-rings, or similar.

Once the geometry is fixed, the “locally ringed topological spaces” with respect to it will be topological spaces $X$ with a structure sheaf exhibited by a suitable functor $\mathcal{G} \to Sh_{(\infty,1)}(X)$. That’s supposed to be the same as a suitable functor $\mathcal{T} \to Sh_{(\infty,1)}(X)$ for $\mathcal{T}$ the corresponding pregeometry.

Here formal duals of $C^\infty$-rings would form the pregeometry, and formal duals of simplicial (“derived”) $C^\infty$-rings the geometry.

In David Spivak’s work, essentially, up to some technical subtlety, the pregeometry $\mathcal{T}$ is taken to be the category of manifolds or Cartesian spaces, hence formal duals of free $C^\infty$-rings. Again, not that there is any problem with that, I am just making a remark on this from the point of view of what one might call “derived synthetic differential geometry”.

Of course, one has to be careful about a certain iteration of concepts here:

a $C^\infty$-ring it itself not unlike a “0-truncated structured $\infty$-topos” on the pregeometry of duals to free $C^\infty$-rings: namely a product preserving functor

$CartSp \to Sh(*) = Set \,.$

And it is this perspective that David Spivak varies, by paassing instead to functors

$CartSp \to Sh_{(\infty,1)}(*) = sSet$

($\infty-C^\infty$-rings) and then further to locally $\infty$-$C^\infty$-ringed topological spaces $X$

$CartSp \to Sh_{(\infty,1)}(X)$

that constitute the $\infty$-category $dDiff$ (or $dMan$) of “derived smooth manifolds”.

(I am just sketching the rough outline, saying this with all technical qualifiers fully spelled out takes a bit longer, of course.)

Now, from the perspective of derived synthetic differential geometry, what one could do to get the $\infty$-stacks with notions of infinitesimals that I mentioned at the beginning is to take this now as a new category of test spaces , i.e. to pass now from $\infty$-Lie groupoids/$\infty$-stacks on $Diff$ to $\infty$-stacks on $dDiff$!

There is some iteration process going on, where we bootstrap a notion of generalized spaces in terms of $\infty$-stacks on test spaces that are themselved generalized spaces obtained as $\infty$-stacks on other test spaces. It can get a bit confusing. Which is why I thought this is worth making a remark about.

Posted by: Urs Schreiber on February 4, 2010 11:51 AM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

Derived? meaning??

Posted by: jim stasheff on February 4, 2010 1:26 PM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

Derived? meaning??

It’s supposed to remind you of derived category. But – on the other hand – not too much!

In “derived geometry” derived is meant in the good $(\infty,1)$-categorical sense, whereas in “derived category” it refers to the decategorification to a 1-category of an $(\infty,1)$-category.

Notably, in “derived algebraic geometry” one passes from the category of rings to that of simplicial rings . But by the monoidal Dold-Kan correspondence (Schwede-Schipley), the category of simplicial rings is equivalent to the category of non-negatively graded chain dg-algebras, hence to monoids in the category of chain complexes, whose homotopy category is the derived category of modules. That’s one path to connect the two usages of the word “derived” here.

I wasn’t asked when the term “derived geometry” was established, and I think it not a good choice of term at all. But I can try to say what made people come up with it.

It is in fact a bit ironic, I think, since as far as I am aware the term “derived” dates from a time when it was unclear what exactly, conceptually, one did, and all one knew was that one could somehow derive one category from another (abelian) category. The point of “derived geometry” as understood today is the opposite, now one actually knows what it is going on. Therefore I would have thought the right term to use would have been

• higher geometry

• $\infty$-categorical geometry

or the like. But then, considering $\infty$-stacks on 1-categorical sites such as that of manifolds (as we mention above) or the Nisnevich site of schemes (as in motivic cohomology) is also already higher $\infty$-categorical geometry, but is not called “derived geometry”, because the underlying site is still 1-categorical.

The way the term is used currently is that we say derived geometry when the generalized spaces under consideration are $\infty$-stacks on a genuinely $\infty$-categorical site.

Or better, I think you can read Structured Spaces as follows:

take an ordinary site over which you modeled your ordinary geometry and regard it as a pregeometry $\mathcal{T}$. Then the geometric envelope of this pregeometry is a geometry $\mathcal{G}$. And this may be thought of as encoding the derived geometry of $\mathcal{T}$.

For instance when we start with the pregeometry $\mathcal{T} = (CRing^{fin})^{op}$ that underlies ordinary algebraic geometry, then the corresponding geometric envelope is the site of simplicial commutative rings. This is Structured Spaces, prop 4.2.3.

So I suppose a good way to say it would be that

Passing to derived geometry means passing from a pregeometry to its geometric envelope, in the sense of Structured Spaces .

Posted by: Urs Schreiber on February 4, 2010 2:03 PM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

I think this may be slightly incorrect: the name derived category is not to emphasise being derived as a whole from abelian; it stems from earlier terminology. Namely, the derived category is the natural setup to study and naturally redefine the derived functors in homological algebra (classical derived functors were an earlier notion!), without reference to a particular resolution, its objects are already “prepared”, “derived” and hence suitable for derived functors (for classical derived functors, the derived is from having a connected sequence of cohomology like functors derived from a single functor; in addition the business of differential in complexes may have had to do as well with the choice of terminology by Cartan and Eilenberg). The same purpose and logic is for the derived scheme and derived geometry. Not for being derived from usual geometry, but for being suitable for derived/homotopically correct replacements of non-exact constructions. In the main original purpose of derived geometry, that is, constructing correct moduli spaces, one first constructs some ambient space of cocycles, where the moduli live; then one needs to replace single out by setting up equations/equalizers the right ones. This means taking a limit. To make it correct you take homotopy limit, or derive from the left. Then one sees that there are automorphisms, which are THE reason why Grothendieck (in dispair of abundance of automorphisms in problems he studied) introduced stacks, what means deriving from the right: quotienting by automorphisms, by taking homotopy cofibers. Now derived moduli space obtained is indeed obtained by deriving in the sense of derived functors. And once there all operations are correct derived, internally in this world. Just like total derived functors between the derived categories.

Posted by: Zoran Skoda on February 5, 2010 4:44 AM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

Thanks, Zoran, for amplifying derived functors.

I feel that what you say does not contradict what I said, but rather supports it. Derived functors go between homotopy categories (total ones at least), hence in particular between derived categories. Whether the notion of category or of functor is more fundamental depends maybe on perspective, but I take the point that historically derived functors were considered before derived categories.

Still, my same comment as to the usage of the word “derived” applies. As you say yourself:

its objects are already “prepared”, “derived” and hence suitable for derived functors (for classical derived functors, the derived is from having a connected sequence of cohomology like functors derived from a single functor;

So “derived” originally just meant something like “we can obtain one thing from another thing by deriving the latter from the former in some way”. It’s hard to think of a word more vague for an operation that that it “derives” something. I think it is clear that this comes from from a situation where it was not fully clear what happens conceptually when functors are “derived”.

Posted by: Urs Schreiber on February 5, 2010 10:19 AM | Permalink | Reply to this

Re: Derived Synthetic Differential Geometry

I’ll vote for Zoran’s treatment since it emphasizes the homotopy/homological aspects. To paraphrase Thom,
a word that means everything means nothing. Or
WS Gilbert: when everyone is somebody, then noones anybody ;-)

Posted by: jim stasheff on February 5, 2010 1:52 PM | Permalink | Reply to this

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