Transfinite limits in topos theory

TL;DR

This paper introduces a novel construction of an enlarged coherent site in topos theory, preserving cohomology and enabling transfinite compositions, inspired by Bhatt and Scholze's pro-etale topology.

Contribution

It develops a new method to enlarge coherent sites that maintains cohomological properties and introduces a weak form of algebraic closure within topos theory.

Findings

Cohomology of bounded below complexes is preserved in the enlargement.

Transfinite compositions of epimorphisms remain epimorphisms in the enlarged topos.

A weak analog of algebraic closure is established in the new construction.

Abstract

For a coherent site we construct a canonically associated enlarged coherent site, such that cohomology of bounded below complexes is preserved by the enlargement. In the topos associated to the enlarged site transfinite compositions of epimorphisms are epimorphisms and a weak analog of the concept of the algebraic closure exists. The construction is a variant of the work of Bhatt and Scholze on the pro-etale topology.