Glad to see you all are keeping busy.

A very imprecise question: Is there a "theory" of elliptic complexes (like coherent sheaves) for compact smooth manifolds with nice functorial properties and whose K-groups are K-homology? Maybe ideally it would have pushforward which could be computed by first factoring into a closed embedding and a submersion. For submersion use integration. For embedding factor through tubular neighborhood and use some sort of cutoff (I do not expect this theory to be local), its vague but still...

I would very much like to have a version of Atiyah Index theorem which resmbels formally GRR. In the sense that both sides are different decategorifications of the same category (coherent sheaves). Maybe analytic K-homology is the answer?

@SaalHardali Not exactly what you're looking for, but Fredholm complexes seem relevant

This might be the answer. Basically the question can be simplified to just, what is K-homology a decategorification of? It it happens to be something which makes sense for any reasonable compact hausdroff space all the better if it just makes sense for manifolds so be it. But i would really like to have something like that no matter its scope.

Actually this might not be really motivated from GRR, im not sure whats K-homology of a scheme is either.

What's nice in GRR is that you have a relative version for any map. In Atiyah Singer i think theres a version for smooth submersion and even on this i'm not sure about...

I guess I just need to think about it more and identify exactly what bothers me in the formulation of AS

Do you know Segal's description of K-homology using "linear combination of points with coefficients in vector spaces"?

Yeah. That's nice, but it's a bit of a different direction.

Basically the thing i want to be able to say is that coherent sheaves on a complex manifolds are a particular case of elliptic complexes

And have everything about coherent sheaves all functors and stuff be extended in some way to elliptic complexes in such a way being a coherent sheaf will just be another condition on an elliptic complex.

(of course you'd need complexes of elliptic complexes for coherent sheaves which are not vector bundles)

but you get the idea...

Then hopefully have a relative atyiah singer for elliptic complexes which will generalize relative version of GRR

But for really straightforward reasons because one theory embeds into the other.

(modulo GAGA stuff)

Maybe even have a verdier duality for elliptic complexes strictly generalizing serre duality

@JonathanBeardsley I was actually looking at your paper on the enriched Grothendieck construction, and I was intrigued by something you wrote in the introduction. Is the only obstruction to generalizing it the fact that V-Cat is not V-enriched?

What do you "get" if you just re-run your argument with an enriched base? Is it something that you can wrap your head around?

@DenisNardin I think I got a piece of the puzzle in case youre interested. So it turns out that from the point of view of analytic K theory the algebra of compact operators is the same as a point. Now recall that for a compact manifold the algebra of pseudodifferential operators is an extension of the algebra of continuous functions on S*M with compact operators (i'm not completely sure yet exactly what variant of pseudodifferential symbols I should use but i'm sure there is one s.t. its true)

Now use that the suspension of SM is thom space of TM which is stably the n-th suspension of the spanier whitehead dual of M. So knowing the K homology of M is the same as knowing the K-theory of S*M which by the above is the same as analytic K theory of pseudodifferential operators. Now notice that an elliptic complex gives a well defined fredholm complex by taking sobolev sections for large s (because they are invertible modulo compact operators).

Now if (E,d) is the elliptic complex and h is the null homotpy modulo compact operators (also called the parametrix). Then i think the K_0 class corresponding to this complex via duality is d+h inside the K_1 of S*M (i.e. the torsion of the null homotopy which should be independent of h).

So basically the relation between elliptic complexes and K-homology is pretty weird due to the fact that Th(TM) is only the suspension of SM.

@DenisNardin, is that right? I think you're referring to the \emph{codomain fibration} $S^2\to S$, but there the Cartesian arrows are just pullbacks. These would correspond to the maps $(f,f^\sharp)$ where $f^\sharp$ is an isomorphism. I want to allow $f^\sharp$ to be any morphism, as in the category of schemes.

@DavidSpivak Sorry, I shouldn't answer in the evening. Complete brainfart. Let me reread your question

Ok, that's the total category of the Grothendieck op-fibration classified by the functor $s\mapsto S_{/s}$

In this paper: arxiv.org/abs/1409.2165 we call it the dual fibration to the Grothendieck fibration $\mathrm{Ar}\,S\to S$

Assuming you mean $B\times_{B'}E'$, of course :)

Ugh wait

Ah ops, that's not the op-fibration, that's the fiberwise op

I'm terrible distinguishing left and right

It's the opposite category of the total category of the op-fibration

You can see that this has a projection to $S$ sending $[E→B]$ to $B$ whose fibers are exactly $(S_{/B})^{op}$, and this is a Grothendieck fibration more or less tautologically

Ugh I guess this works only if you assume that the morphism $E'\times_{B'}B→E$ is over $B$, which I've kind of taken for granted