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A higher categorical approach to Giraud's non-abelian cohomology

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posted on 28.03.2022, 12:44 authored by Alexander Campbell
This thesis continues the program of Ross Street and his collaborators to develop a theory of non-abelian cohomology with higher categories as the coeffcient objects. The main goal of this thesis is to show how this theory can be extended to recover Giraud's non-abelian cohomology of degree 2, thereby addressing an open problem posed by Street. The definition of non-abelian cohomology that is adopted in this thesis is one due to Grothendieck, which takes higher stacks as the coeffcient objects; the cohomology is the higher category of global sections of the higher stack. The two approaches of Street and Grothendieck are compared and it is shown how they may be reconciled. The central argument depends on a generalisation of Lawvere's construction for associated sheaves, which yields the 2-stack of gerbes over a site as an associated 2-stack; the stack of liens (or \bands") is the 1-stack truncation thereof. Much of the work is dedicated to showing how the coherence theory of tricategories, supplemented by results of three-dimensional monad theory and enriched model category theory, provides a practicable model of the tricategory of indexed bicategories over a site,in whose context the theory can be developed. This tricategory is shown to be triequivalent to a full sub-Gray-category of the Gray-category of indexed 2-categories over the site, thus permitting the tricategorical analogues of limits, colimits, image factorisation systems, and Grothendieck's plus construction to be modelled by strict constructions of Gray-enriched category theory.

History

Table of Contents

Chapter 1. Introduction -- Chapter 2. The central argument -- Chapter 3. A coherent to 2-stacks -- Chapter 4. Non-abelian cohomology -- Chapter 5. Torsors, gerbes and Giraud's H2 -- Chapter 6 Conclusion.

Notes

Theoretical thesis. Includes bibliographical references

Awarding Institution

Macquarie University

Degree Type

Thesis PhD

Degree

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

Department, Centre or School

Department of Mathematics

Year of Award

2016

Principal Supervisor

Street Ross

Rights

Copyright Alexander Campbell 2016. Copyright disclaimer: http://mq.edu.au/library/copyright

Language

English

Extent

1 online resource (vi, 129 pages : illustrations)

Former Identifiers

mq:70195 http://hdl.handle.net/1959.14/1261186