A higher categorical approach to Giraud's non-abelian cohomology
thesisposted on 2022-03-28, 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.