Dissertations, Theses, and Capstone Projects

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Mahmoud Zeinalian

Committee Members

Martin Bendersky

Thomas Tradler

Subject Categories

Geometry and Topology

Keywords

diffeological spaces, category theory

Abstract

Finite dimensional smooth manifolds have been studied for hundreds of years, and a massive theory has been built around them. However, modern mathematicians and physicists are commonly dealing with objects outside the purview of classical differential geometry, such as orbifolds and loop spaces. Diffeology is a new framework for dealing with such generalized smooth spaces. This theory (whose development started in earnest in the 1980s) has started to catch on amongst the wider mathematical community, thanks to its simplicity and power, but it is not the only approach to dealing with generalized smooth spaces. Higher topos theory is another such framework, considerably more abstract and based heavily on categorical and homotopical techniques. In this dissertation, these two points of view are combined. We draw a bridge between these frameworks by using a cofibrant replacement functor of Dugger's to embed diffeological spaces into simplicial presheaves in a homotopically correct way. From this we prove that the theory of bundles between these two frameworks agree. We then port over the powerful tools of higher topos theory, such as the shape operator, to obtain new results in diffeology. As our main result, we obtain a short exact sequence exhibiting the obstruction to the Cech-de Rham isomorphism for diffeological spaces in all dimensions, building on an analogous result of Patrick Iglesias-Zemmour's in dimension 1.

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