2023/12/22 by Giorgio Trentinaglia, Trentinaglia, Giorgio
Mathematics · Medicine · #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2312.14908
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.