Classifying Tangent Structures using Weil Algebras
Poon Leung · Theory and applications of categories · 2017
At the heart of differential geometry is the construction of the tangent bundle of a manifold.There are various abstractions of this construction, and of particular interest here is that of Tangent Structures.Tangent Structure is defined via giving an underlying category M and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behaviour of T in the category End(M).However, this axiomatic definition at first seems somewhat disjoint from other approaches in differential geometry.The aim of this paper is to present a perspective that addresses this issue.More specifically, this paper highlights a very explicit relationship between the axiomatic definition of Tangent Structure and the Weil algebras (which have a well established place in differential geometry).Thanks to Steve Lack for his immense support and to the referee for his/her many helpful suggestions.