The p -order of topological triangulated categories

Abstract

The p‐order of a triangulated category is an invariant that measures ‘how strongly’ p annihilates objects of the form Y/p. In this paper, we show that the p‐order of a topological triangulated category is at least p−1; here we call a triangulated category topological if it admits a model as a stable cofibration category. Our main new tools are enrichments of cofibration categories by Δ‐sets; in particular, we generalize the theory of ‘framings’ (or ‘cosimplicial resolutions’) from model categories to cofibration categories.

The n -order of algebraic triangulated categories

“…Section 3 contains our main result: for every prime p, the p-order of the p-local stable homotopy category is at most p − 1. In particular, the p-local stable homotopy category is not algebraic for any prime p; this is folklore for p = 2, but seems to be a new result for odd primes p. In a companion paper we show that for every prime p, the p-order of every topological triangulated category is at least p − 1 [9,Thm. 5.3]; so the p-order of the p-local stable homotopy category is exactly p − 1.…”

“…Since the stable homotopy category is a topological triangulated category, its p-order and that of any subcategory is at least p − 1 by Schwede [9,Theorem 5.3]. By Theorem 3.1, the p-order of SH c (p) , and hence of every triangulated category which contains it, is at most p − 1.…”

“…Since the stable homotopy category is a topological triangulated category, its p-order and that of any subcategory is at least p − 1 by [9,Thm. 5.3].…”

“…The results of this paper were announced in [8] and the contents of this paper and its companion [9] were originally combined as a single paper (arXiv:1201.0899); the referee convinced the author to divide up the results into two separate papers.…”

Highern-angulations from local rings

“…Moreover, they proved that this particular triangulated category is neither algebraic nor topological. For a discussion on algebraic and topological triangulated categories, see, for example, [14][15][16].…”