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created etale (infinity,1)-site
You work with sCAlg and then at one place with infinity-CAlg. Is this intentional ?
I had chosen to make the definition that $\infty CAlg_k$ denotes the $\infty$-category presented by the model structure on $sCAlg_k$. I have now put that into a Definition-environment so that it is less easy to miss.
The point is here to remind us by the notation that simplicial algebras are just a model for an intrinsic notion of $\infty$-algebra over an $(\infty,1)$-algebraic theory.
Do you claim that all obvious variants of (infinite,1)-categorifications of etale site are equivalent ?
You see, for example in the holomorphic setup the concept of simplicial Stein site is for example not that obvious from the point of view of Stein 1-site (the work of Finnur Larusson mentioned recently uses this concept.
Do you claim that all obvious variants of (infinite,1)-categorifications of etale site are equivalent ?
I am reproducing specifically the $(\infty,1)$-site that Toën-Vezossi and Lurie consider as the $\infty$-categorification of the standard étale site.
But notice that Jacob Lurie gives a characterization that shows one way in which this is the “right” $\infty$-version:
His Structured Spaces, prop. 4.3.15 shows that if you take the ordinary etale site and regard it as a “pregeometry” in his sense, then the $(\infty,1)$-etale site as defined here is the corresponding “geometry” in his sense.
That’s the way he formalizes what “derived geometry” means: you take an ordinary site, regard it as a “pregeometry” and then pass to the “geometric envelope”.
That does not mean that there cannot be other useful/interesting $\infty$-versions of the etale 1-site. But it gives a certain justification for why one might want to use that one.
for example in the holomorphic setup the concept of simplicial Stein site is for example not that obvious from the point of view of Stein 1-site (the work of Finnur Larusson mentioned recently uses this concept
I have to run now. Could you make a brief note on this simplicial site in an $n$Lab entry? Then I’ll look at it in an hour or so.
I added the detailed definition and statement of how the etale $\infty$-site is formally the “derived geometry” (= geometric envelope) of the ordinary etale site: derived etale geometry.
Thanks!
When you write “category of homotopy components”, do you mean the homotopy category?
Then that makes it a simplicial site.
It is simplicial site. The words homotopy category as you know have many meanings. Here I mean just the path components in the sense of paths in compact-open category (unless I am halucinating this late, what is likely). In other words the $hom(X,Y)$ will be $\pi_0(Map(X,Y))$.
When we talk Stein there is an interesting abstract of a past Koncevič’s talk in Miami:
Any (Wein)stein manifold X can be contracted to a singular Lagrangian submanifold L, e.g. by the gradient flow of a plurisubharmonic function. I’ll argue that L carries a homotopy cosheaf of dg-categories of finite type, whose global section is the wrapped Fukaya category of X. In the case when X is a surface with boundary, we obtain a description of F(X) as a homotopy colimit of a finite diagram of representation categories of quivers of series A. Further examples include Riemann-Hilbert correspondence for irregular holonomic D-modules (via Stokes structures), and dg-algebras associated with Legendrian links.
In other words the $hom(X,Y)$ will be $\pi_0(Map(X,Y))$.
Good, that’s the homotopy category of a simplicial category. In any case, that fits with simplicial site.
Of course. But the nonobvious thing is to take the enrichment given by (edit erased holomorphic here) singular simplicial set of a (compactly-open topologized) mapping space of holomorphic maps and the topology given by maps which have to be deformed into biholomorphic onto Stein (and these are jointly surjective). This is nontrivial modification from the usual Stein site.
By the way, in the definition with $\pi_0$ above $Map(X,Y)$ is not a simplicial category (instead it is a topological space), $Map_\Delta(X,Y)$ is (for notation see simplicial Stein site). Not a big deal of course.
I just checked Finnur’s article linked at simplicial Stein site, and he doesn’t use holomorphic simplices in the holomorphic mapping space (with compact-open topology), but all simplices. I’ve changed the definition at simplicial Stein site accordingly.
Right David. I just seconds before your post came to a moment of sanity: once one gets to $Map(X,Y)$ in compact open topology, it does not make sense to talk about the holomorphic singular simplices into it as the target is not a finite dimensional complex manifold. On the other hand, there is another construction which does relate the whole site with the business of singular simplices I vaguely remember.
Edit: crossover with Zoran’s post immediately preceeding this one.
There may be a little confusion with the holomorphic singular set of a complex manifold - this latter is defined using the standard affine simplices. $\Delta^n_a$ is a hyperplane in $\mathbb{C}^{n+1}$, and the usual simplicial identities are what you might expect from taking $0,1 \in \mathbb{C} \simeq \Delta^1_a$ as the ’boundary’ of the 1-simplex. This isn’t defined for the mapping space. But I could imagine that you could take the $n$-simplices of $Map_\Delta(X,Y)$ to be maps $X \times \Delta^n_a \to Y$. I couldn’t guarantee this gave the same simplicial set unless $X$ and $Y$ were cofibrant and fibrant or whatever the analogues are (Stein/elliptic/etc).
Something of the sort, thanks.
okay, so do we agree now that this is a simplicial site as defined there?
I confirmed even still in #10 that it is a simplicial site. Why are you asking again ? P.S. I never disagreed with this part – why would somebody call it simplicial Stein site if it were not a simplicial site ? (on the other hand, the homotopy category is defined directly in terms of $Map$ which are defined in the entry, not in termy of $Map_\Delta$ which gives enrichement. Of course it boils down to the same thing at the end.)
I was just trying to make sure that we are really talking about the same definition, just because you used that unusual term for “homotopy category”. People could well be using the term “simplicial site” with different precise meaning in mind. I was just trying to make sure.
Homotopy category is overloaded term, so I find it more clear saying for such a variant decriptively in two words – set of components up to homotopy or set of path components. In 13 I said that it is of course about homotopy category question. The point which I am emphasising all the time and which made me sleeping one hour later last night in order to communicate to you, and which is still without your reaction has nothing to do with variants of infinity-site but of nonobvious nature of concrete recipe for categorification. I mean going to speak about covers by Stein manifolds in the sense of mapings from Stein manifolds into the target which are homotopic to biholomorphic mappings onto open Stein subsets which themselves cover in usual sense; as well as starting with holomorphic inner homs and then doing singular sset in compact/open topology on that holomorphic inner hom. I do not know if you can give a recipe for categorifying Stein site to get this (or something provably equivalent) and the categorification of etale site with the same recipe.
and which is still without your reaction has
That’s not true. I told you the recipe for how to do it for the etale site. But I haven’t done the analogous computation for the Stein site, so I cannot tell you about the result there.
My understanding is that both 1- and infinity,1-geometries (and all in between) are obtained from single pregeometry, not that 1-geometry determines a pregeometry (hence indirectly higher geometries), in general. You are claiming even more that there is a computable answer in some sense, for the composition.
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