Elimination of quotients in various localisations of premodels into models

TL;DR

This paper unifies various localization schemes in category theory, provides a general construction from premodels to models, introduces techniques to analyze universal properties, and presents a quotient elimination process to organize complex relational data in higher-dimensional structures.

Contribution

It unifies multiple localization frameworks, offers a general construction method, introduces assessment techniques for universal properties, and develops a quotient elimination process for higher-dimensional models.

Findings

Unified various localization schemes in category theory.

Developed a general construction from premodels to models.

Introduced a quotient elimination process for organizing relational data.

Abstract

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega$-spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allows one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(\omega,n)$-categories, weak $\omega$-groupoids and higher moduli stacks.