Higher Vector Bundles
Matias del Hoyo, Giorgio Trentinaglia
TL;DR
The paper develops intrinsic geometric models for representations up to homotopy of higher Lie groupoids by introducing higher vector bundles endowed with cleavages. It constructs a semi-direct product that yields an equivalence between nonnegative representations up to homotopy and higher vector bundles with a normal weakly flat cleavage, establishing a higher Grothendieck correspondence. This framework extends the Dold-Kan correspondence and the Gracia-Saz–Mehta VB-2-term story to higher contexts and provides a concrete model for homotopy colimits via a relative nerve-like construction. It lays groundwork for derived versions, Morita invariance, and tensor product structures, linking higher representation theory with intrinsic simplicial geometry and potential applications in geometry and physics.
Abstract
We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between representations up to homotopy and simplicial vector bundles endowed with a cleavage. Our result is an incarnation of the Higher Grothendieck Correspondence, it can be seen as a relative Dold-Kan Theorem and extends earlier work of Gracia-Saz and Mehta on VB-groupoids.