By the way, Dylan, @EricPeterson pointed out to me that you all were able to prove that HF_p can't be an E_n Thom spectrum for n>2. That's super cool (and mysterious...)!
thanks! and that's only for odd primes (HF_2 is an E_3-Thom spectrum)... which somehow makes it even more mysterious
and HF_2 is an E_3-Thom spectrum in some sneaky way, but it means we can, in theory, write HF_2 in terms of "E_3 generators and relations" in some manageable way according to the cell structure on HP^{infty}. It'd be neat to know if there was some analogously manageable way to write HF_p as an E_3 algebra in terms of generators and relations, even though we can't do the Thom-trick.
well if you're building your algebra using elements of the picard group, then any diagram will factor through its group completion, and then you'd have a thom spectrum. But you could imagine trying to write down E_n-lax-monoidal functors to the category of modules itself. The trouble is that, when things are group-like we have this neat trick: E_n-maps are the same as maps between deloopings, which we sometimes can write down. And we're good at writing down maps between spaces.
if things aren't group like, then the deloopings are (\infty, n)-categories. And I don't think we're as good at writing down maps between (\infty,n)-categories as we are at writing down maps between spaces
sorry, the above is hard to parse,in the first part I mean: If C is E_n-monoidal, then any functor C--->Pic(Sp) will factor through a group-like E_n-space, since the target is a group-like E_n-space. So we don't get anything new as long as we're thinking about the "generators" as coming from the picard groupoid. But if instead we moved to diagrams that landed just in Sp (or even the full subcategory on the invertible objects instead of the maximal subgroupoid), then we might get something new.
in the second part I mean: To build E_n-maps X-->Pic(Sp) when X is group-like can be reduced to building a map out B^nX which is often accessible because we know how to build maps between spaces. But to build lax E_n-monoidal functors C--->Sp we would have the look at the (\infty, n)-category B^nC and study appropriately lax functors B^nC--->B^nSp of (\infty, n)-categories. But I think it's much less common that we have access to 'categorical' deloopings of E_n-monoidal categories, and even
less common that we can produce maps out of (infty,n)-categories easily. (in the space case we have like... obstruction theory, and cell decompositions... but I imagine a 'cell decomposition' of an (infty,n)-category is harder to come by.)
Does the category of constructible sheaves on a real analytic manifold care about the smooth structure? The definition of constructible certainly involves the smooth structure (through Whitney conditions) but intuitively it looks to me like it would be very weird if this category wasn't a homeomorphism invariant...
I was trying to read this paper (arxiv.org/abs/1409.0501) by Francis, Ayala and Tanaka which develops big machines for general stratified spaces when i realized i have no idea to what extent is a "constructible" sheaf a topological notion...
From the contents of the paper it seems reasonable that the smooth structure shouldn't be relevant for defining the category of constructible sheaves but classically it looks far from obvious that this is indeed the case
@SaalHardali their definition of "stratified space" is not exactly the same as others, since it's meant for different purposes (namely the encoding of $(\infty,n)$-categories and related notions through manifold topology). i'm not sure what's going on in other definitions, but here once you know the stratified topological space (just meaning a topological space equipped with a continuous map to a poset) then that determines what "constructible" means.
(and i believe that for their purposes, "constructible" always just means "constructible with respect to the chosen stratification")
@DylanWilson there's still an E_3-cell structure that's analogous that you get by calculating the E_3-TAQ, and you're still attaching one cell in each degree congruent to 1 mod 4. when you take homology you get your generator tau_0 in degree 1 and a relation in degree 4 ([tau_0, tau_0] = 0) and i suspect that lots of the relations are of the form "all these browder brackets need to please go away"
maybe that's the wrong relation and it should be the restriction from the Lie algebra
hmm but i guess there's an earlier question here, namely: "given a topological manifold, do its different smooth structures accommodate different stratifications?" i don't know the answer, but i feel like it should be "yes" (in the AFT formalism) because stratified spaces have a pretty subtle notion of smooth structure, namely "conical smoothness", and surely by "constructible sheaf on a smooth manifold" we mean "constructible with respect to a conically smooth stratification"
@AaronMazel-Gee That's what I gathered but then I wonder if any of the classical theory (Kashiwara-Schapira's tome) can be said in this language or these are just 2 completely different meanings for the word "constructible"
If you take for example the definition in Kashiwara-Schapira (which is used and referenced a lot) it relies heavily on sub analytic sets which depend on the analytic structure nevermind smooth structure.
There's this thing here which smooth can always be upgraded to analytic uniquely upto some $\epsilon$ which I never knew the value of
I see. Do you know the value of $\epsilon$ for which one can say "a smooth manifold can be given an analytic structure uniquely upto $\epsilon$" without lying
I feel this is kind of relevant here, at least in determining if the category of constructible sheaves (in K-S) is diffeomorphism invariant
@AaronMazel-Gee If so then let me give you another question of this flavour which troubles me: Lets say two sheaves are non-characteristic for each other if their singular supports intersects at the zero section. Is the notion of being non-characteristic homeomorphism invariant?
I mean if there's justice in the world it has to be.
@DylanWilson well, so one thing that might fail is that the attaching map for the 5-cell is present in the space Pic(free E_3-algebra with p=0) but doesn't lift to Pic(S)
say if our prime is biiig and so that Pic(S_{(p)}) looks like an Eilenberg-Mac Lane spectrum in that range?
Here's a historical question. If $G$ and $H$ are 1-groupoids, then $\mathrm{Map}(BG,BH) \approx B\mathrm{Fun}(G,H)$. The question is: who originally proved/observed this? Or maybe more usefully: is there someone's name which ought to be attached to this result (because they very likely stated it very differently than I just did).
