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Affine geometric spaces in tangent categories

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R. F. Blute, G. S. H. Cruttwell, and R. B. B. Lucyshyn-Wright

We continue the program of *structural differential geometry* that
begins with the notion of a tangent category, an axiomatization of
structural aspects of the tangent functor on the category of smooth
manifolds. In classical geometry, having an affine structure on a
manifold is equivalent to having a flat torsion-free connection on its
tangent bundle. This equivalence allows us to define a category of affine
objects associated to a tangent category and we show that the resulting
category is also a tangent category, as are several related categories.
As a consequence of some of these ideas we also give two new
characterizations of flat torsion-free connections.
We also consider 2-categorical structure associated to the category of
tangent categories and demonstrate that assignment of the tangent
category of affine objects to a tangent category induces a 2-comonad.

Keywords:
Tangent categories, affine manifolds, connections

2010 MSC:
18D99, 53A15, 53B05, 53C05, 18F15

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 15, pp 405-437.

Published 2019-04-29.

http://www.tac.mta.ca/tac/volumes/34/15/34-15.pdf

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