Configuration complexes and a variant of Cathelineau's complex in weight 3

In this paper we consider the Grassmannian complex of projective configurations in weight 2 and 3, and Cathelineau's infinitesimal polylogarithmic complexes. Our main result is a morphism of complexes between the Grassmannian complex and the associated infinitesimal polylogarithmic complex. In order to establish this connection we introduce an $F$-vector space $\beta^D_2(F)$, which is an intermediate structure between a $\varmathbb{Z}$-module $\mathcal{B}_2(F)$ (scissors congruence group for $F$) and Cathelineau's $F$-vector space $\beta_2(F)$ which is an infinitesimal version of it. The structure of $\beta^D_2(F)$ is also infinitesimal but it has the advantage of satisfying similar functional equations as the group $\mathcal{B}_2(F)$. We put this in a complex to form a variant of Cathelineau's infinitesimal complex for weight 2.