Fibrations and Koszul duality in locally Cartesian localisations

TL;DR

This paper develops a framework for understanding fibrations in locally Cartesian localisations of presentable infinity-categories, introducing a Koszul duality classification and applying it to various important examples.

Contribution

It establishes a right proper model structure with all morphisms cofibrations for these localisations and provides a Koszul duality classification of fibrations, including criteria for nullification functors.

Findings

Fibrations in these localisations can be classified via Koszul duality.

The framework applies to nullification functors with non-trivial homotopical content.

Examples include local systems, A1-covering spaces, and loop space modules.

Abstract

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.