Differentiating $L_\infty$ groupoids

TL;DR

This paper proves that differentiating Lie n-groupoids yields a tangent complex that is representable and carries a Lie n-algebroid structure, advancing the understanding of their differential geometry.

Contribution

It demonstrates that the presheaf from differentiating Lie n-groupoids is representable by the tangent complex, which naturally inherits a Lie n-algebroid structure.

Findings

The presheaf from differentiating Lie n-groupoids is representable.

The tangent complex of a Lie n-groupoid has a Lie n-algebroid structure.

Differentiation yields a tangent complex with rich geometric structure.

Abstract

Differentiating an Lie $n$-groupoid via the differential-geometric fat point a priori only yielads a presheaf of graded manifolds. In this article we prove that this presheaf is representable by the tangent complex of the Lie $n$-groupoid. As an immediate consequence we obtain that the tangent complex carries the structure of a Lie $n$-algebroid.