# Cubical VS Simplicial Manifold and the De Rham theorem

Let $\boldsymbol{\Delta}$ be the simplex category. Let ${\Box}$ be the cube category. I denote with $\Delta[n]$ the standard $n$-simplex and with $\Delta_{geo}[n]$ its geometric realization. On the other hand there I will denote with $\Box[n]$ the standard $n$-cube and with $\Box_{geo}[n]$ its geometric realization. In the paper "[Dp] Simplicial de Rham cohomology and characteristic classes of flat bundles" written by J.L. Dupont there are some interesting facts involving simplicial manifolds and their cohomology. I wonder if similar statements are true as well for "cubical" manifolds. Let $\Bbbk$ be a field of characteristic zero.

For a set $X$ I denote with $(X)^{\Bbbk}$ the $\Bbbk$-module of maps from $X$ to $\Bbbk$. Consider $\Delta[\bullet]$ as a cosimplicial simplicial set, hence $(\Delta[\bullet])^{\Bbbk}$ is a simplicial cosimplicial $\Bbbk$-module. For each cosimplicial module $(\Delta[n])^{\Bbbk}$ I denote with $C_{n}$ its associated cochain graded module and with $NC_{n}$ its normalize cochain graded module. Hence $C_{\bullet}$, $NC_{\bullet}$ are both simplicial cochain graded modules. On the other hand let $\Omega[n]$ be the differential graded algebra of polynomial $\Bbbk$ forms on the geometric n-simplex, hence $\Omega[\bullet]$ is a simplicial differential graded module. In [Dp] there is a proof of the following fact:

(a)For each $n$ there are two cochain maps $E_{n}\: : \: NC_{n}\to \Omega[n]$, $F_{n}\: : \: \Omega[n]\to NC_{n}$ and a cochain homotopy $s_{n}\: : \: \Omega[n]\to \Omega[n]$ such that $F_{n}\circ E_{n}=Id$ and $E_{n}\circ F_{n}$ is homotopic via $s_{n}$ to the identity.

There is an explicit combinatorial formula for the maps $(E_{\bullet}, F_{\bullet},s_{\bullet})$ mentioned above. Let $A^{\bullet,\bullet}$ be a cosimplicial non negatively cochain graded module. Here the first bullet denotes the cosimplicial degree. For each $n\in \boldsymbol{\Delta}$ let $N(A^{n,\bullet})$ be its normalize complex, then $N(A^{\bullet,\bullet})$ is a bicomplex and we denote with $Tot(N(A))$ the associated total complex. As a corollary of $(a)$ we have the following:

Consider the functor $\Omega[\bullet]\otimes A^{\bullet}\: : \: \Delta^{op}\times \Delta\to dg-Mod$. Then the triple $(E_{\bullet}, F_{\bullet},s_{\bullet})$ induces two chain maps $E\: : \: Tot(N(A))\to \int_{\boldsymbol{\Delta}}\Omega[\bullet]\otimes A^{\bullet}, \quad F\: : \: \int_{\boldsymbol{\Delta}}\Omega[\bullet]\otimes A^{\bullet}\to Tot(N(A))$ and an homotopy $s\: : \: \int_{\boldsymbol{\Delta}}\Omega[\bullet]\otimes A^{\bullet}\to \int_{\boldsymbol{\Delta}}\Omega[\bullet]\otimes A^{\bullet}$ such that $F\circ E=Id$ and $E\circ F$ is homotopic via $s$ to the identity. In particular the coend $\int_{\boldsymbol{\Delta}}\Omega[\bullet]\otimes A^{\bullet}$ is quasi isomorphic to $Tot(N(A)$.

b) Let $M_{\bullet}$ be a (real or complex) smooth simplicial manifold. We denote with $B^{\bullet,\bullet}$ its de Rham algebra, i.e $B^{n,q}$ is the module of differential forms of degree $q$ on $M_{n}$. In particular it is a cosimplicial differential commutative algebra. Let $||M_{\bullet}||$ be the fat geometric realization of $M_{\bullet}$. In [Dp] there is a proof of the following

$H^{\bullet}(||M||)\cong H^{\bullet}(Tot(N(B)))$

Finally, here my questions: are the results for a) and b) true if I replace the $\boldsymbol{\Delta}$ with $\Box$, if I replace (co)simplicial **** with (co)cubical ****?

1)For a). The cubical version of the statements in a) are easy to formulate. Let $\Omega^{\Box}[n]$ be the differential graded algebra of polynomial differentials forms on the n-cube. Then for a cocubical cochain module $A^{\bullet,\bullet}$ we have a functor $\Omega^{\Box}[\bullet]\otimes A^{\bullet}\: : \: \Box^{op}\times \Box\to dg-Mod$. Is it true that the coend $\int_{\Box}\Omega[\bullet]\otimes A^{\bullet}$ is quasi isomorphic to $Tot(N(A)$? Is it possible to find such a map explicit?(maybe by using the triple (E,F,s) above...).

2) Is the statement b) true as well for cubical manifolds? Should I add some conditions to $M_{\bullet}$?

It seems to me that the answer for 1) is Yes, but I don't see how to prove that. I don't have any idea about 2). My only worries is that the normalization procedure for cubical modules may have a different homotopy type. This does not happen with simplicial objects.