Nerve theorems for Lie Theory

Abstract: Tangent categories provide an abstract characterization of the tangent bundle construction for differential geometry. They provide a convenient setting for the semantics of differentiable programming languages and have made some analogies to differential calculus concrete (e.g. McCarthy's functor calculus from homotopy theory). In this talk we consider a pair of 'nerve' constructions in algebraic topology -- Grothendieck's nerve construction that embeds the category of small categories into the category of simplicial sets, and Kan's nerve/realization adjunction between simplicial sets and topological spaces -- and how they arise in Lie theory when using a tangent-categorical perspective. In doing so we find a more conceptual approach to the differentiation of Lie groupoids to Lie algebroids and the Euler-Poincare/Euler-Weinstein-Martinez formulation of classical mechanics on a Lie algebroid/groupoid. Along the way we shall consider an alternative definition of Lie algebroids based on Eduardo Martinez's 'canonical involution' of a Lie algebroid.