The n-Category Café

Thanks for catching that Toby.

Fixed.

Urs - that’s very interesting. A couple of comments. First, in the fiber sequence for a TX action, was the base supposed to be B(TX), the classifying space of the tangent algebroid (which is the quotient of X by the formal neighborhood of the diagonal) rather than TX?

Second, as you imply, you can define D-modules in any context where you have a notion of tangent complex, and a notion of stable sheaf (eg stabilization of stacks). I know tangent complexes can be defined in huge generality (see eg Toen-Vezzosi’s HAG2), eg when you start with a category of commutative ring objects in some symmetric monoidal ∞-category. I would assume you have tangent complexes in your smooth context as well? or equivalently, that you can talk about formal neighborhoods of the diagonal, or about de Rham complexes? (Sorry, I assume this is what you’re explaining here, but I didn’t follow the details)

Third, what you end up defining sounds to me (probably naively) like representations of the path groupoid, which are not what I would think of as analogs of D-modules — representations of the path (oo)-groupoid are local systems, which are very special D-modules. D-modules can have interesting singularities, ie are not locally constant - their topological avatar are constructible sheaves. Constructible sheaves with respect to a fixed stratification are representations of the sub(oo-)groupoid of the path groupoid that preserves this stratification (and there are various refinements you are aware of in papers of Treumann and Wise etc). In any case I don’t think the path groupoid is the groupoid formally integrating the tangent algebroid (ie for which modules are the same as those for the tangent algebroid) - the path groupoid is the analog of a Lie group with Lie algebra TX, but Lie algebra modules are equivalent to representations of the formal group, not the (a) corresponding group, unless you impose strong finiteness (which kills the singularities in D-modules).. Well anyway it’s possible for your purposes local systems suffice? they do in the homotopy category of spaces (but not in the category of smooth manifolds).

Thanks for your reply.

I would assume you have tangent complexes in your smooth context as well?

Okay, so that was the crux of the question here. Let me try to say this more in detail:

For $X$ a manifold let $\Pi(X)$ be its smooth path $\infty$-groupoid regarded as a simplicial sheaf.

Let $C^\infty(-)$ be the map that reads in a sheaf on $Diff$ and spits out a product-preserving copresheaf on cartesian spaces, i.e. a Moerdijk-Reyes smooth algebra.

Then applying $C^\infty(-)$ degreewise yields a cosimplicial copresheaf

$$C^\infty(\Pi(X)) \,.$$

Under the dual Dold-Kan correspondence this maps to the complex that computes smooth singular cohomology of $X$.

Since $\Pi(X)$ is in fact a simplicial concrete sheaf, it makes sense to speak of open neighbourhoods of degenarate elements and we may also consider the analogous construction but with all functions replaced by just local functions (germs at totally degenerate cells). That yields the cosimplicial copresheaf

$$C^\infty_{loc}(\Pi(X))$$

that I mentioned. Under dual Dold-Kan both yield cochain complexes that should be equivalent to the deRham complex of $X$.

Analogously, for $G$ a Lie group and $\mathbf{B} G$ the corresponding simplicial sheaf, we get that the cosimplicial copresheaf

$$C^\infty_{loc}(\mathbf{B}G)$$

is the one that computes local Lie group cohomology and is hence, under dual Dold-Kan, weakly equivalent to the Chevalley-Eilenberg complex $CE(Lie(G))$.

So based on this what I was suggesting is this:

to any simplicial sheaf $A$ assign the cosimplicial copresheaf $C^\infty_{loc}(A)$ of local Moerdijk-Reyes function algebras in the above sense.

That should be a model for the $L_\infty$-algebroid corresponding to $A$.

Now, by the properties of Moerdijk-Reyes algebras it should be true that this assignment $C^\infty_{loc}(-)$ sends pullbacks to pushouts.

$$C^\infty(X_1 \times_Y X_2) \simeq C^\infty(X_1) \otimes_{C^\infty(Y)} C^\infty(X_2) \,.$$

Given this I suggested to view the category of modules for $\Omega^\bullet(X)$ as the under-category of cosimplicial Moerdijk-Reyes algebras under $C^\infty_{loc}(X)$.

Generally, this might suggest to assign to any simplicial sheaf $A$ the under-category $Q(A) := C^\infty_{loc}(A)/CoSCoSh$.

I am playing with the thought that this $A \mapsto Q(A)$ produces a good general notion of “geometric $\infty$-functions”.

There seems to be an interesting variant on that “bare-bones geometric function theory” given by over-categories

$$C(A) := \mathbf{H}/A$$

as described here:

this assignment yields functorial pull-push through spans. But not through spans-of-spans. But, unless I am mixed up, we do get the higher degree functorial pull-push if instead of the plain over-category $\mathbf{H}/A$ we take spans in that:

$$C : A \mapsto Spans(\mathbf{H}/A) \,.$$

Here one needs to say what “Spans(\mathbf{H}/A)” means for $A$ something an $\infty$-groupoid (in the case that $\mathbf{H} = \infty Grpd$). Of course Jacob Lurie did that in OTCTFT, around page 57, where it’s called $Fam_1(A)$.

I am thinking that the assignment

$$C : A \mapsto Fam_1(A)$$

with $Fam_1$ not just necessarily interpreted in $\mathbf{H} = \infty Grpd$ should yield a “geometric function theory” in the more-groupoidification-like sense of the discussion here which is not only functorial with respect to pull-push along spans, but also along spans-of-spans.

(Draw the obvious diagram on paper to see what I am thinking of, or else wait until I post a graphic tomorrow).

More generally then it should be true that the assignment

$$C : A \mapsto Fam_n(A)$$

yields something for which pull-push extends to an $(n+1)$-functor on $(n+1)$-fold spans-of-spans.

If this is correct and I am not mixed up, that should allow to extend the groupoidification/”geometric $\infty$-function theory” pull-push propagation to higher dimensions.

Of course such a higher dimensional abstract-nonsense-path-integral thing is precisely what Freed-Hopkins-Lurie-Teleman hint at in their latest, and using precisely these $Fam_n$-constructions. So I have to emphasize that what I just said, in the case anyone actually read it, means a use of the notion of the $Fam_n(-)$ construction very different in flavor to that in FHLT, even if motivated by the same goal.

Makes me wonder. But it’s too late today, I’ll continue wondering tomorrow morning.