2021/05/26 by Andrew W. Macpherson, Macpherson, Andrew W.
Mathematics · Physics and Astronomy · #18N40 55U35 18N60 18F25 14F42 #Algebra over a field #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Cartesian closed category #Cartesian coordinate system #Categorical variable #Category Theory (math.CT) #Computer science #Discrete mathematics #Duality (order theory) #Extension (predicate logic) #FOS: Mathematics #Functor #Geometry #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Mathematics #Model category #Morphism #Nonlinear Waves and Solitons #Pure mathematics #Simple (philosophy) #Subcategory #math.AT #math.CT #msc:14F42 #msc:18F25 #msc:18N40 #msc:18N60 #msc:55U35
paper · pdf · doi:10.48550/arxiv.2105.12316
20 pages. v2: Examples substantially expanded, added Koszul duality statements. The title of this paper was previously "Locally Cartesian localisations and the higher Thomason model structure." Keywords: model infinity category, right proper, nullification, A1 homotopy, plus construction, groupoid completion, Koszul duality
openalex publication_date 2021/05/26 · arxiv created 2021/08/12 · arxiv updated 2021/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I show that any locally Cartesian left localisation of a presentable infinity-category admits a right proper model structure in which all morphisms are cofibrations, and obtain a Koszul duality classification of its fibrations. By a simple criterion in terms of generators for a localisation to be locally Cartesian, this applies to any nullification functor. In particular, it includes examples with non-trivial "homotopical content." I further describe, and provide examples from, the set of fibrations in three contexts: the higher categorical Thomason model structure of Mazel-Gee, where fibrations are local systems; Morel-Voevodsky A1-localisation, where they are a higher analogue of A1-covering spaces; and the Quillen plus construction, where they are related to loop space modules trivialised over the universal acyclic extension.