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An introduction to differential bundles

Banff International Research Station for Mathematical Innovation and Discovery

This tutorial will show how algebraic structure in tangent categories can capture geometric differential structure by considering the relationship between vector bundles and differential bundles in the category of smooth manifolds. Vector bundles are fibered vector spaces that are also fibre bundles, so they are not essentially algebraic in the sense of Freyd. Differential bundles, however, are coalgebras for the weak comonad induced by the vertical lift on the tangent bundle satisfing a universal property. We will begin by showing that Cockett and Cruttwell's original characterization of a differential bundle is equivalent to the current definition. Then, we will show that the functor from vector bundles to differential bundles is an isomorphism of categories.

Mathematics; Category Theory; Homological Algebra; Algebraic Topology; Algebraic Structures

video/mp4

Author affiliation: University of Calgary

2023-10-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/86376

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/