Title: Global to local to global: differentiation and atlases in category theory
Abstract: In many different setups throughout mathematics, differentiation is used to analyze a globally complicated situation locally. Conversely, small objects are glued to form a larger object using atlases. Instead of studying these concepts in one specific setup (e.g. smooth manifolds), we use category theory to study the constructions without specifying what the underlying objects are. This talk is meant to be a broad overview over some of these constructions in category theory.
We study categorical constructions that allow differentiation and atlases, in particular tangent categories and restriction categories. We will define tangent categories and see that they unify different historical approaches to derivatives. In tangent categories many constructions from classical geometry still work. We will in particular see that dimensions, vector bundles and Lie groups work very similar to their classical versions.
In the second half of the talk study atlases in join restriction categories, instructions how to glue objects together using maps that are only defined on a subset of their domain. We use atlases to construct principal bundles, objects which locally look like the Cartesian product of a space and a group.
This will be the final seminar before my thesis defense.
