Schreiber
de Rham theorem for ∞-Lie groupoids

Idea

In the context of a smooth (∞,1)-topos $H$ the differential Quillen adjunction

$${\Pi }^{\inf }:H\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}^{\inf }$$\Pi^{inf} : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat}^{inf}

implies that for $X$ fibrant or cofibrant (or both, of course) the flat differential cohomology $H({\Pi }^{\inf }(X),B^{n}A)$ of $X$ coindices with the ordinary cohomology $H(X,(B^{n}A{)}_{\mathrm{flat}}^{\inf })$ of $X$ with coefficients in the object $(B^{n}A{)}_{\mathrm{flat}}^{\inf }$.

In the case that $A$ is a contractible abelian Lie group, one finds

• that this coefficient object is equivalent to $B^n{A}_c$, where $A_c$ is $A$ regarded as a geometrically discrete group;

• that flat differential $A$-valued cohomology coincides with $A$-valued deRham cohomology.

So in this case the differential Quillen adjunction restricts to the statement

$$H_{\mathrm{dR}}^n(X,A)\simeq H^n(X,A_c).$$H^n_{dR}(X,A) \simeq H^n(X,A_c) \,.

In the case that the ∞-Lie groupoid $X$ appearing here is just an ordinary manifold this is the ordinary deRham theorem.

Details

Recall the embedding of strict ∞-Lie groupoids into all ∞-Lie groupoids, which allows us to identify an $\mathbb{N}$-graded chain complexes of sheaves of abelian groups $A^{•}\in \mathrm{Sh}(C,{\mathrm{Ch}}_{•}(\mathrm{Ab}))$ on our site $C$ of test spaces with an ∞-Lie groupoid $A\in \mathrm{SSh}(C)$ modeled as a simplicial sheaf on $C$.

Using this identification, for $A\in \mathrm{Sh}(C)\hookrightarrow \mathrm{SSh}(C)$ an abelian group object – an abelian Lie group – its $n$-fold delooping $B^{n}A\in \mathrm{SSh}(C)$ is the Eilenberg-MacLane object given by the chain complex with $A$ concentrated in degree $n\in \mathbb{N}$:

$$B^{n}A\simeq A[n]:=(\cdots \to 0\to A\to 0\to \cdots \to 0).$$\mathbf{B}^n A \simeq A[n] := (\cdots \to 0 \to A \to 0 \to \cdots \to 0) \,.

We shall freely make use of this identification in the following.

Let now and in the following $A\in \mathrm{Sh}(C)\hookrightarrow \mathrm{SSh}(C)$ be an abelian group object in our smooth (∞,1)-topos that is an ordinary abelian Lie group.

Write

$$A_c\in \mathrm{Sh}(C)\hookrightarrow \mathrm{SSh}(C)$$A_c \in Sh(C) \hookrightarrow SSh(C)

for the geometrically discrete version of $A$, which as a sheaf is the constant sheaf of constant maps into $A$.

Proposition

With $A$ and $A_c$ as above, there is a weak equivalence

$$(B^{n}A{)}_{\mathrm{flat}}^{\inf }\stackrel{\simeq }{\to }{B}^n{A}_c$$(\mathbf{B}^n A)_{flat}^{inf} \stackrel{\simeq}{\to} \mathbf{B}^n A_c

in $\mathrm{SSh}(C{)}_{\mathrm{loc}}$.

Proof

Using general facts about combinatorial differential forms with values in abelian groups we find that $(B{A}^n{)}_{\mathrm{flat}}^{\inf }:=\mathrm{SSh}({\Pi }^{\inf }(-),A)$ is given by the flat Deligne complex

$$(B^{n}A{)}_{\mathrm{flat}}^{\inf }=(\cdots 0\to C^{\mathrm{\infty }}(-,A)\stackrel{d\log }{\to }{\Omega }^1(-,A)\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^2(-,A)\stackrel{d_{\mathrm{dR}}}{\to }\cdots \stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^n(-,A)).$$(\mathbf{B}^n A)_{flat}^{inf} = (\cdots 0 \to C^\infty(-,A) \stackrel{d log}{\to} \Omega^1(-,A) \stackrel{d_{dR}}{\to} \Omega^2(-,A) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(-,A) ) \,.

The quasi-isomorphism from this chain complex to

$$B^n{A}_c=(\cdots \to 0\to A_c\to 0\to \cdots \to 0)$$\mathbf{B}^n A_c = (\cdots \to 0 \to A_c \to 0 \to \cdots \to 0)

is a fundamental fact in abelian sheaf cohomology following from the (synthetic?) Poincare lemma?. As discussed at strict ∞-Lie groupoid this quasi-isomorphism of chain complexes of sheaves constitutes a weak equivalence of the corresponding simplicial sheaves under the objectwise Dold-Kan correspondence.

Corollary

For $A$ as above and for $X$ fibrant, infinitesimal flat differential $B^{n}A$-valued cohomology on $X$ is isomorphic to ordinary $A_c$-cohomology.

$$H({\Pi }^{\inf }(X),B^{n}A)\simeq H(X,B^n{A}_{\mathrm{flat}}^{\inf })\simeq H(X,B^n{A}_c)$$H(\Pi^{inf}(X), \mathbf{B}^n A) \simeq H(X,\mathbf{B}^n A_{flat}^{inf}) \simeq H(X, \mathbf{B}^n A_c)

Proof

By assumption all representables are contractible objects. Fibrant objects in $\mathrm{SPSh}(\mathrm{contr}𝕃{)}_{\mathrm{proj}}^{\mathrm{loc}}$ are those that are degreewise Kan complexes and that satisfy descent on all representables. But

• everything in the image of the Dold-Kan correspondence is objectwise a Kan complex,

• and on contractibles the descent condition is trivial

so that we conclude that regarded as objects in $\mathrm{SPSh}(\mathrm{contr}𝕃{)}_{\mathrm{proj}}^{\mathrm{loc}}$ all coefficient objects appearing here are fibrant.

Then for $Y\stackrel{\simeq }{\to }X$ a cover of $X$ with $Y$ cofibrant, we have that ${\Pi }^{\inf }(Y)$ is also cofibrant (as discussed at infinitesimal path ∞-groupoid) and that ${\Pi }^{\inf }(Y)\to {\Pi }^{\inf }(X)$ is a weak equivalence (by the properties discussed there).

Using the infinitesimal differential Quillen adjunction $({\Pi }^{\inf }(-)⊢(-{)}_{\mathrm{flat}}^{\inf })$ and noticing the nature of the (∞,1)-categorical hom-space we have

$$\begin{array}{rl}H(\Pi (X),B^{n}A)& :={\pi }_{0}H(\Pi (X),B^{n}A)\\ & \simeq {\pi }_0{\mathrm{SSh}}_C(\Pi (Y),B^{n}A)\\ & \simeq {\pi }_0{\mathrm{SSh}}_C(Y,(B^{n}A{)}_{\mathrm{flat}}^{\inf })\\ & \simeq {\pi }_{0}H(X,(B^{n}A{)}_{\mathrm{flat}}^{\inf })\\ & =:H(X,(B^{n}A{)}_{\mathrm{flat}}^{\inf })\end{array}$$\begin{aligned} H(\Pi(X), \mathbf{B}^n A) & := \pi_0 \mathbf{H}(\Pi(X), \mathbf{B}^n A) \\ & \simeq \pi_0 SSh_C(\Pi(Y), \mathbf{B}^n A) \\ & \simeq \pi_0 SSh_C(Y, (\mathbf{B}^n A)_{flat}^{inf}) \\ & \simeq \pi_0\mathbf{H}(X, (\mathbf{B}^n A)_{flat}^{inf}) \\ & =: H(X, (\mathbf{B}^n A)_{flat}^{inf}) \end{aligned}

and then using the above proposition

$$\begin{array}{rl}\cdots & \simeq {\pi }_0{\mathrm{SSh}}_C(Y,(B^{n}A{)}_{\mathrm{flat}}^{\inf })\\ & \simeq {\pi }_0{\mathrm{SSh}}_C(Y,B^n{A}_c)\\ & \simeq {\pi }_{0}H(X,B^n{A}_c)\\ & =:H(X,B^n{A}_c)\end{array}.$$\begin{aligned} \cdots & \simeq \pi_0 SSh_C(Y, (\mathbf{B}^n A)_{flat}^{inf}) \\ & \simeq \pi_0 SSh_C(Y, \mathbf{B}^n A_c) \\ & \simeq \pi_0 \mathbf{H}(X, \mathbf{B}^n A_c) \\ & =: H(X, \mathbf{B}^n A_c) \end{aligned} \,.

Corollary (deRham theorem)

For $A$ as above but contractible as a topological space ordinary $A_c$-cohomology coincides with $A$-valued deRham cohomology.

$$H(X,B{A}_{\mathrm{dR}}^n)\simeq H(X,B^n{A}_c)$$H(X,\mathbf{B}A^n_{dR}) \simeq H(X, \mathbf{B}^n A_c)

Proof

Recall that (infinitesimal) nonabelian deRham cohomology of $X$ is the relative cohomology of ${\Pi }^{\inf }(X)$ with respect to the inclusion $X\hookrightarrow {\Pi }^{\inf }(X)$. But for $A$ being contractibe, any $B^{n}A$-cocycle on $X$ is necessarily trivializable, so that inn this case deRham cohomology coincides already with the cohomology of ${\Pi }^{\inf }(X)$. The statement then follows by the above corollary.

References

The deRham theorem on the level of the homotopy category of ∞-stacks is asserted in

• Carlos Simpson, Constantin Teleman, deRham theorem for $\mathrm{\infty }$-stacks (pdf)