Almost mathematics of pointed symmetric monoidal model categories by Smith ideal theory

TL;DR

This paper extends the concept of almost mathematics to symmetric monoidal pointed model categories using Smith ideals, establishing a weak correspondence analogous to Quillen's module theory over non-unital rings.

Contribution

It introduces a novel application of Smith ideals to almost mathematics in the context of symmetric monoidal pointed model categories, generalizing Quillen's bilocalization results.

Findings

Established a weak analogue of Quillen's one-to-one correspondence.

Applied Smith ideal theory to model categories.

Extended almost mathematics to new categorical frameworks.

Abstract

This article is a generalization of a result in Quillen's note ``Module theory over non-unital rings'' giving a one-to-one correspondence between bilocalization of abelian categories of modules and idempotent ideals of the base ring. Faltings; Gabber and Ramero established almost mathematics, the same as Quillen's bilocalization of a category of modules by nil modules. In this paper, by using the theory of Smith ideals mentioned in Hovey and Smith, we consider almost mathematics of symmetric monoidal pointed model categories. We prove a weak analogue of the one-to-one correspondence in Quillen.