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considerably expanded Lie infinity-groupoid. But still stubby.
keep adding stuff to Lie infinity-groupoid, now there are sections on Lie 1- and 2-groups, their depooping and their “flat” objects, smooth nonabelian cohomology, flat Deligne cohomology, flat nonabelian cohomology. Some still a bit stubby, but mainly fully fledged with formal theorems and detailed proofs.
Am working on the entry Lie infinity-groupoid:
have been expanding the section on differential coefficients for Lie-integrated $\infty$-groups, i.e. the discussion of differential refinements of those Lie $\infty$-groups $G$ that arise as the integration $\tau_n \exp(\mathfrak{g})$ of an $L_\infty$-algebra $\mathfrak{g}$.
There is now detailed proof of the statements
that the presheaf $G TrivBund_{flat}$ of trivial $G$-principal $\infty$-bundles with flat $\infty$-connetion is equivalent to the underlying discrete $\infty$-groupoid $\mathbf{\flat} \mathbf{B}G$ of $\mathbf{B}G$
and factors the canonical inclusion $\mathbf{\flat}\mathbf{B}G \to \mathbf{B}G$ as a fibration;
as a corollary one gets a model for the de Rham coefficient object $\mathbf{\flat}_{dR} \mathbf{B}G$ (the coefficient object for nonabelian de Rham cohomology with values in $\mathfrak{g}$).
It is maybe noteworthy that $\mathbf{B}G$ is the concretization (in the sense of concrete presheaf) of $G TrivBund_{flat}$ and $\mathbf{\flat}_{dR} \mathbf{B}G$ is the kernel of this concretization, i.e. the maximally non-concrete sub-object.
Wow, that sounds cool. Sounds like a nonabelian generalization of some famous exact sequence involving de Rham cohomology and Deligne cohomology.
Yeah, it looks like that.
There’s a displayed equation in the $n$Lab entry that didn’t compile…
Thanks for your comment!
Sounds like a nonabelian generalization of some famous exact sequence involving de Rham cohomology and Deligne cohomology.
Yes, it’s closely related.
For the case that the $\infty$-group $G$ is “mildly abelian” (namely braided as a monoidal $\infty$-groupoid, in that two deloopings of it exist) the exact sequence that relates differential cohomology with coefficients in $G$ and de Rham cohomology with coefficients in $G$ is derived here (scroll down just a little bit to the lemma called “differential fiber sequence”).
And it is a direct formal consequence of the fiber sequence that I was discussing above, which I write
$\mathbf{\flat}_{dR} \mathbf{B}G \to \mathbf{\flat} \mathbf{B}G \to \mathbf{B}G$The trick is to read “$\mathbf{\flat}$” as “flat”: so $\mathbf{\flat}\mathbf{B}G$ is the coefficient object for flat $G$ $\infty$-bundles (it’s the discrete $\infty$-groupoid underlying the Lie $\infty$-groupoid $\mathbf{B}G$. Then by definition the homotopy fiber $\mathbf{\flat}_{dR} \mathbf{B}G$ is the coefficient for $G$-valued de Rham cohomology (and the definition justifies itself by reducing to what one expectes to see in the suitable special cases).
In the fully nonabelian case, i.e. when $G$ is only once deloopable or when instead of $\mathbf{B}G$ we use any $\infty$-Lie groupoid $A$ there is also an exact sequence for differential $G$-cohomology. This involves nonabelian de Rham cohomology with coefficients in the object which I write $\Omega \mathbf{\Pi}_{dR} \mathbf{B}G$, where $\Omega$ denotes looping, of course, and $\mathbf{\Pi}_{dR} \mathbf{B}G$ is the homotopy cofiber of the constant path inclusion $\mathbf{B}G \to \mathbf{\Pi}\mathbf{B}G$.
That’s the topic of the section Differential cohomology with non-groupal coefficients.
All this is completely “formal” in that it involves only abstract operations in an oo-connected (oo,1)-topos. The discussion at Lie infinity-groupoid that I mentioned above is about what all this abstract stuff boils down to concretely when realized in the $(\infty,1)$-topos $\infty LieGrpd$.
started a section The Lie-integrated universal principal $\infty$-bundle
added a section on descent for strict oo-Lie groupoids.
am starting sections on integration of an oo-Lie algebra cocycle to a cocycle of $\infty$-Lie groupoids; and on the oo-Chern-Weil homomorphism. But stubby for the moment.
started a new section titled simplicial de Rham complex whose punchline is supposed to be this:
the literature knows two standard models for the simplicial de Rham complex of a simplicial manifold $X_\bullet$, with a fiber integration map constituting a quasi-isomorphism between them.
On the other hand, I happen to have two natural models (in terms of complexes of sheaves, as described in the entry) for the intrinsic de Rham coefficient object $\mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}$, and a fiber integration map constitutes a weak equivalence between them.
The claim is: the cocycle oo-groupoids of the two models for the simplicial de Rham complex are precisely the hom-complexes of morphisms of simplicial presheaves
$X_\bullet \to \mathbf{\flat}_{dR} \mathbf{B}^n \mathbb{R}$into the given two models, respectively, and the quasi-isomorphism is that induced from the weak equivalence between these.
I spelled out some details about how to prove this in the section linked to above and indicated how to continue. But have to interrupt now.
This sounds like it would be worth submitting as a brief note for publication. Not every publication needs to be earth shattering (few are). I obviously don’t understand the details, but it sounds like a nice little result.
In another thread I had a few days ago announced a computation of the intrinsic de Rham cohomology of $\mathbf{B}^n U(1)$ in the $(\infty,1)$-topos of $\infty$-Lie groupoids. The first versions of what I had typed out had however been quite rough.
I have now refined this to what I think is a rather detailed proof. I moved this to the section oo-Lie groupoids – Intrinsic de Rham cohomology of Bn U(1).
As I try to indicate in the proof, using the projective model structure on simplicial presheaves there is an easy naive version where one ignores the need to pass to a cofibrant resolutioin of $\mathbf{B}^n U(1)$. The nontrivial work is required in demonstrating that the correct computation of maps out of a cofibrant resolution does reduce to this naive version after all.
By the relation of cohomology in simplicial presheaves to ordinary abelian sheaf hypercohomology (as for instance in Jardine’s lecture notes) there should be an alternative computation where we don’t resolve $\mathbf{B}^n U(1)$ but use instead an injective resolution of the complex of sheaves $(\Omega^1(-) \stackrel{d_{dR}}{\to} \to \cdots \to \Omega^n_{cl}(-))$. First I tried to compute it this way, but then I wasn’t sure how to handle some details.
On the other hand, the proof that I have given now (in as far as it is correct) has the advantage over this would-be alternative proof that its structure works also for nonabelian coefficients.
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