In this talk I want to report about ongoing work in collaboration with Matias del Hoyo which opens a new perspective on the notion of representation up to homotopy introduced by Abad and Crainic. Our research aims at clarifying the relationship between higher vector bundles (a special class of simplicial vector bundles) and representations up to homotopy of Lie n-groupoids. In the spirit of the classical Dold-Kan correspondence, we have found a simple combinatorial/ geometric explanation for the equations which underlie the notion of representation up to homotopy, leading to a functorial correspondence between higher vector bundles with higher connections ("cleavages") and representations up to homotopy. The correspondence between connections on VB-groupoids and two-term representations up to homotopy pointed out by Gracia-Saz and R. Mehta is a very special case of (and served as motivation to) our result, which encompasses the arbitrary n-term case (n possibly infinite). We expect our point of view to shed new light on various aspects of the theory, for instance, on tensor products.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |