The n-Category Café

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

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.

I feel like rather a wet blanket today:
1. A knowingly misleading headline isn’t “good”.
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.

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…)

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.

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.

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?

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