Geodesic
Affine geometric spaces in tangent categories
Theory and applications of categories, Tome 34 (2019), pp. 405-437.

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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.
Publié le :
Classification : 18D99, 53A15, 53B05, 53C05, 18F15
Keywords: Tangent categories, affine manifolds, connections 
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     author = {R. F. Blute and G. S. H. Cruttwell and R. B. B. Lucyshyn-Wright},
     title = {Affine geometric spaces in tangent categories},
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     pages = {405--437},
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     volume = {34},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a14/}
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R. F. Blute; G. S. H. Cruttwell; R. B. B. Lucyshyn-Wright. Affine geometric spaces in tangent categories. Theory and applications of categories, Tome 34 (2019), pp. 405-437. http://geodesic.mathdoc.fr/item/TAC_2019_34_a14/