## TOPOLOGICAL TRIANGULATED CATEGORIES

### BibTeX

@MISC{Schwede_topologicaltriangulated,

author = {Stefan Schwede},

title = {TOPOLOGICAL TRIANGULATED CATEGORIES},

year = {}

}

### OpenURL

### Abstract

Many triangulated categories arise from chain complexes in an additive or abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying additive categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, and formalize this in Definition 1.4 via homotopy categories of stable cofibration categories. The purpose of this paper is to explain some systematic differences between algebraic and topological triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. The precise statements use the n-order of a triangulated category, for n a natural number (see Definition 2.1). The n-order is a non-negative integer (or infinity), and it measures, roughly speaking, ‘how strongly ’ the relation n · Y/n = 0 holds for the objects Y in a given triangulated category (where Y/n denotes a cone of multiplication by n on Y). Our main results are: • The n-order of every algebraic triangulated category is infinite (Theorem 3.3). • For every prime p, the p-order of every topological triangulated category is at least p − 1 (Theorem 8.2). • For every prime p, the p-order of the p-local stable homotopy category is exactly p − 1