Topological triangulated categories

TL;DR

This paper distinguishes algebraic and topological triangulated categories using the 'n-order' concept, showing that topological ones have finite n-order while algebraic ones have infinite n-order, with implications for the p-local stable homotopy category.

Contribution

It introduces the 'n-order' invariant to differentiate algebraic and topological triangulated categories and develops foundational tools for cofibration categories.

Findings

Algebraic triangulated categories have infinite n-order.

Topological triangulated categories have finite p-order, at least p-1.

The p-local stable homotopy category has p-order exactly p-1.

Abstract

In this paper we explain certain systematic differences between algebraic and topological triangulated categories. A triangulated category is algebraic if it admits a differential graded model, and topological if it admits a model in the form of a stable cofibration category. The precise statements use the 'n-order' of a triangulated category, for a natural number n. The n-order is a non-negative integer (or infinity) and measures `how strongly' n annihilates objects of the form Y/n. We show the following results: the n-order of an algebraic triangulated category is infinite; for every prime p, the p-order of a topological triangulated category is at least p-1; the p-order of the p-local stable homotopy category is exactly p-1. In particular, the p-local stable homotopy category is not algebraic for any prime p. As a tool we develop certain foundations about enrichments of cofibration categories by Delta-sets; in particular we generalize the theory of `framings' (or `cosimplicial resolutions') from model categories to cofibration categories.