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Classifying tangent structures using Weil algebras

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Poon Leung

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).

Keywords:
Tangent Structure, Weil algebra

2010 MSC:
18D99,53A99

*Theory and Applications of Categories,*
Vol. 32, 2017,
No. 9, pp 286-337.

Published 2017-02-15.

http://www.tac.mta.ca/tac/volumes/32/9/32-09.pdf

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