Motivic toposes

TL;DR

This paper proposes a framework using topos theory to analyze Weil-type cohomology theories and motives, aiming to construct classifying toposes and a unifying syntactic triangulated category.

Contribution

It introduces a novel topos-theoretic framework for motives and cohomology theories, including the construction of a syntactic triangulated category linking to classical cohomologies.

Findings

Framework based on atomic two-valued toposes and homogeneous models

Preliminary results towards classifying toposes of Weil-type cohomologies

Construction of a syntactic triangulated category

Abstract

We present a research programme aimed at constructing classifying toposes of Weil-type cohomology theories and associated categories of motives, and introduce a number of notions and preliminary results already obtained in this direction. In order to analyze the properties of Weil-type cohomology theories and their relations, we propose a framework based on atomic two-valued toposes and homogeneous models. Lastly, we construct a syntactic triangulated category whose dual maps to the derived categories of all the usual cohomology theories.