Algebraic versus topological triangulated categories

TL;DR

This paper compares algebraic and topological triangulated categories, highlighting systematic differences mainly related to torsion phenomena, which vanish rationally, and discusses their distinct origins and properties.

Contribution

It clarifies the fundamental differences between algebraic and topological triangulated categories, especially regarding torsion-related properties and their implications.

Findings

Algebraic categories satisfy certain properties not shared by topological ones.

Differences are primarily torsion phenomena, absent rationally.

Topological categories originate from non-additive contexts, unlike algebraic ones.

Abstract

The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called `algebraic' because they originate from abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the source categories are usually very `non-additive' before passing to homotopy classes of morphisms. Because of their origin I refer to these examples as `topological triangulated categories'. In these extended talk notes I explain some systematic differences between these two kinds of triangulated categories. There are certain properties -- defined entirely in terms of the triangulated structure -- which hold in all algebraic examples, but which fail in some topological ones. These differences are all torsion phenomena, and rationally there is no difference between algebraic and topological triangulated categories.