The Adjoint Representation of a Higher Lie Groupoid

TL;DR

This paper generalizes the concept of the adjoint representation from Lie groupoids to higher Lie groupoids, establishing a well-defined, unique representation up to homotopy via simplicial vector bundles.

Contribution

It introduces a construction of the adjoint representation for higher Lie groupoids, extending classical notions and proving its independence from choices in the splitting process.

Findings

The adjoint representation of higher Lie groupoids is a well-defined representation up to homotopy.

Existence and uniqueness follow from a general simplicial vector bundle theory.

The construction is independent of the cleavage choice in splitting higher vector bundles.

Abstract

We extend the standard construction of the adjoint representation of a Lie groupoid to the case of an arbitrary higher Lie groupoid. As for a Lie groupoid, the adjoint representation of a higher Lie groupoid turns out to be a representation up to homotopy which is well defined up to isomorphism. Its existence and uniqueness are immediate consequences of a more general result in the theory of simplicial vector bundles: the representation up to homotopy obtained by splitting a higher vector bundle by means of a cleavage is, to within isomorphism, independent of the choice of the cleavage.