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I have been advising Herman Stel on his master thesis, which is due out in a few days. I thought it would be nice to have an nLab entry on the topic of the thesis, and so I started one: function algebras on infinity-stacks.
For $T$ any abelian Lawvere theory, we establish a simplicial Quillen adjunction between model category structures on cosimplicial $T$-algebras and on simplicial presheaves over duals of $T$-algebras. We find mild general conditions under which this descends to the local model structure that models $\infty$-stacks over duals of $T$-algebras. In these cases the Quillen adjunction models small objects relative to a choice of a small full subcategory $C \subset T Alg^{op}$ of the localization
$\mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )$of the $(\infty,1)$-topos of $(\infty,1)$-sheaves over duals of $T$-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical $T$-line object. In as far as objects of $\mathbf{H}$ have the interpretation of ∞-Lie groupoids the objects of $\mathbf{L}$ have the interpretatin of ∞-Lie algebroids.
For the special case where $T$ is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that $T$ is the theory of smooth algebras ($C^\infty$-rings) we obtain a refinement of this to the context of synthetic differential geometry.
As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable $\infty$-stack of infinitesimal paths of a manifold. This construction is an $\infty$-categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the $\infty$-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic $\infty$-vector bundles with flat connection.
The entry is of course as yet incomplete, as you will see.
From the page:
T the theory of ordinary commutative algebras over a field $k$ and $J$ the xyz-topology. In this case the adjunction is that considered in (Toën).
this isn’t related to my fghi-topology is it? :P
this isn’t related to my fghi-topology is it? :P
It arose in the same spirit. But it is meant to be the fpqc-topology.
But the thing is, the only aspect of the topology that matters for the whole argument to go through is that it must be such that the Cech-cohomology of functions on covers vanishes in positive degree and coincides with that of the base in degree 0.
And in fact it seems there is an argument that in degree greater than 2 the cover-cohomology vanishes on general grounds, so that only degree 0 and 1 need to be checked.
Now this work is for abelian Lawvere theory, and I do not know what it exactly means and how essential it is. I see you are using a model structure in this setup. In any case, usually the homotopic constructions extend beyond abelian. For example for operadic context,
“present a general construction of the derived category of an algebra over an operad and establish its invariance properties” (The first sentence from the abstract). One could expect that one can enhance this to infinity category rather than just a “derived category”. I will open a nlab entry for this now: derived category of an algebra over an operad to record the reference.
Abelian Lawvere theory means that it includes the theory of abelian groups. I.e. that an algebra over it has an underlying abelian group.
More on the rest later when I have a minute.
Okay, here I am back with a minute of spare time:
I guess you are asking whether the modl structure on cosimplicial T-algebras that is described in the entry might be equivalent or related to a model structure on algebras over some resolution of the operad corresponding to $T$?
I don’t know. (Would we need operad in cosimplicial sets? ) I’d tend to think that it is another question which is more relevant, but maybe I am wrong. But here is what I tend to think:
one shouldn’t try to think of that model structure on cosimplicial algebras in the entry as something awefully intrinsic, algebraically.. Its whole purpose is rather that it happens to provide a model for the full sub-(oo,1)-catgeory of oo-stacks on duals of T-algebras on those that are R-local, i.e. those such that homming maps into them that induces isos in R-cohomology produces an equivalence. (Where $R$ is the canonical $T$-line object).
As Toen amplifies in his article, that the discussion here is based on, one could in the case where $T$ is the theory of ordinary commutative algebras in principle find models that achieve the same in terms of $E_\infty$-algebras or other structures. But I have not tought about what such other models might be like when we allow for $T$ an arbitrary abelian Lawvere theory. We might get things like $C^\infty-E_\infty$-algebras (i.e. smooth $E_\infty$-algebras), but I don’t know.
On the other hand, what I think is more important as far as homotopy-algebras is concerned is this:
as I try to indicate in the last paragraph of the entry currently, ultimately the full $(\infty,1)$-categorical version of the setup described there uses not Lawvere theories, but (infinity,1)-algbraic theories $T$. This would be fully “derived” $T$-geometry.
So in the full setup, one should do the following: Let $C \subset T Alg^{op}$ be a small full sub(oo,1)-category of the opposite of that of algebras over an $(\infty,1)$-algebraic theory $T$, equip it with the structure of an $(\infty,1)$-site and consider the $(\infty,1)$-topos $\mathbf{H} := Sh_{(\infty,1)}(C)$ of derived $T$-stacks.
Then find models for the reflective sub-$(\infty,1)$-category $\mathbf{L} \hookrightarrow \mathbf{H}$ of the $R$-local objects, where $R$ is again the canonical $T$-line object.
We expect that such a model can again given by cosimplicial $\infty$-$T$-algebras.
In fact, for $T$ the ordinary algebraic theory of ordinary commutative rings , but regarded as an $(\infty,1)$-theory, its algebras are simplicial algebras and the above yields $\infty$-stacks over duals of simplicial algebras as in ordinary derived algebraic geometry. Ben-Zvi and Nadler have looked at Toen’s setup in this context and considered the inclusion of cosimplicial $\infty$-$T$-algebras into these stacks (in their article on Loop spaces and Differential Forms). These are now cosimplicial-simplicial algebras! Which under Dold-Kan are equivalent to unbounded dg-algebras. The duals of that one might call “derived $\infty$-Lie algebroids”. This is where things like the BV-BRSTcomplex lives.
One should eventually try to generalize this from $T$ the ordinary theory of ordinary algebras regarded as an $(\infty,1)$-algebraic theory to $T$ general Lawvere theory regarded as an $(\infty,1)$-algebraic theory (as an intermediate step to allowing $T$ a fully general $\infty$-theory). I regard what the entry currently describes as a warmup for that derived case.
Interesting! Are you also saying that the complicial case (in the sense of Toen, working with model categories for the unbounded complexes) can be fully (including the model structure) be accounted for by cosimplicial/simplicial two step trick ? Toen was saying that the usual intuition/conjectures for concrete constructions in (derived) algebraic geometry often fail when one goes to complicial derived geometry.
Ah, thanks, for reminding me of that use of the word. Yes, it is this sense of “complicial” that I meant.
I believe one should think of it this may, which makes it appear quite natural:
in full generality, we should be looking at things modeled by simplicial presheaves over duals of simplicial algebras
$X : [\Delta^{op}, T Alg] \to [\Delta^{op}, sSet]$This models “derived stacks”. A “function algebra” is now a simplicial $T$-algebra, so as we take degreewise function algebras on simplicial presheaves, we get a cosimplicial simplicial algebra
$\mathcal{O}(X) \in [\Delta, [\Delta^{op}, T Alg]] \,.$Under the Dold-Kan correspondence, this is equivalently a non-negtaively graded cochain complex of non-negatively graded chain complexes
$\cdots \in Ch^\bullet_+(Ch_\bullet^+(Ab))$which is equivalently non-negative cochain complexes of non-positive cochain complexes
$\cdots \simeq Ch^\bullet_+(Ch^\bullet_-(Ab))$The model structure should be such that this is equivalent to the corresponding total complex. But since these double-complexes live in the off-diagonal quadrant, their total complex will be in general unbounded chain complexes. Roughly, the “derived”d riections of the function algebras go towards negative degree, while the “categorical” directions go in positive degree.
For the BV-BRST complex: the ghosts are tangents to higher morphisms of oo-stacks, so they go in positve degrees. The anti-fields are higher graded components in the function algebras, so they go in negative degree.
And, yes, working over the “derived” site of duals of simplicial algebras is strikingly different from working over the site of duals of ordinary algebras, even when ooking at oo-stacks. This difference is most strikingly highlighted by the fact that over simplicial algebras, an ordinary sheaf of sets may have nontrivial loop space objects, something that can never happen if we regard a sheaf of sets over a 1-categorical site.
I have included now into function algebras on infinity-stacks the full details of
the proof of the existence of the claimed model category structure on cosimplicial $T$-algebras;
the proof that for all split hypercovers $Y \to X$ on a subcanonical site we have that $H^n(\mathcal{O}(X)) \to H^n(\mathcal{O}(Y))$ is generally an isomorphism for $n \neq 1$.
the submitted version of Herman Stel’s thesis is now available as a pdf file here.
at function algebras on infinity-stacks I added technical details on the $(\mathcal{O} \dashv Spec)$-adjunction for $\infty$-stacks over a site of duals of simplicial algebras over a field of characteristic 0, following Ben-Zvi/Nadler.
This is in the section function algebras – In derived geometry.
Well, in fact so far (as in BZ/N) this is just abstract nonsese, as it just describes the $(\infty,1)$-Yoneda extension of the inclusion of bounded into unbounded dg-algebras. I am lacking currently a more abstract characterization of this adjunction along the lines of the cohomology-localization theorem that is stated in the first part of the entry for situations over site of plain $T$-algebras. I still need to think about this.
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