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Tangent bundles, monoidal theories, and Weil algebras

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thesis
posted on 28.03.2022, 10:53 authored by Poon Leung
The construction of the tangent bundle of a manifold lies at the very foundations of differential geometry. There are various approaches to characterise the tangent bundle, and two such approaches are through Synthetic Differential Geometry (SDG) and Tangent Structures (in the sense of Cockett-Cruttwell). Here, we shall give a different perspective, that Tangent Structure can be viewed as a model of an appropriate theory. This theory arises as a certain full subcategory Weil1 of the category Weil of all Weil algebras. The connection between Weil algebras and SDG is well established, but their connection to Tangent Structure is not evident. In this thesis, we shall exhibit Weil1 as the universal tangent structure and in fact the axioms of tangent structure actually form a presentation for Weil1. We shall then continue by describing how this perspective allows us to extend this theory in a canonical manner.

History

Table of Contents

1. Introduction -- 2. Weil algebras and graphs -- 3. The category Weil1 -- the category Weil∞ -- 5. Concluding remarks.

Notes

Theoretical thesis. Bibliography: pages 96-97

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Degree

PhD, Macquarie University, Faculty of Science and Engineering, Department of Mathematics

Department, Centre or School

Department of Mathematics

Year of Award

2017

Principal Supervisor

Stephen Lack

Additional Supervisor 1

Richard Garner

Rights

Copyright Poon Leung 2017. Copyright disclaimer: http://mq.edu.au/library/copyright

Language

English

Extent

1 online resource (viii, 97 pages)

Former Identifiers

mq:70142 http://hdl.handle.net/1959.14/1260669