Differentiating $L_\infty$ groupoids -- Part I

Differentiating an $n$-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 $n$-groupoid. As an immediate consequence we obtain that the tangent complex carries the structure of a Lie $n$-algebroid.