A Presheaf Interpretation of the Generalized Freyd Conjecture

We give a generalized version of the Freyd conjecture and a way to think about a possible proof. The essential point is to describe an elementary formal reduction of the question that holds in any triangulated category. There are no new results, but at least one known example drops out very easily. In algebraic topology, the generating hypothesis, or Freyd conjecture, is a long-standing conjecture about the structure of the stable homotopy category. It was initially formulated in 1965 [Freyd, 1966] and remains open to this day. Because the original conjecture has proved difficult to analyze, recent work has turned to studying similar conjectures in categories that share many properties with the stable homotopy category in hopes of further understanding the types of categories in which such a conclusion holds. In this note, we state a version of the generating hypothesis for an arbitrary triangulated category and analyze conditions under which this hypothesis holds. We emphasize that we impose no additional conditions on our triangulated categories, so that our results show which formal properties of a category imply the Freyd conjecture. Therefore our results give conceptual insight into the kind of category in which the generating hypothesis can be expected to hold. 1. The generalized Freyd conjecture Let T be a triangulated category and write [X, Y ] for the abelian group of maps X → Y in T . Let B be a (small) full subcategory of T closed under its translation (or shift) functor Σ and let C be the thick full subcategory of T that is generated by B; write ι : B → C for the inclusion. For emphasis, we often write B(X, Y ) = [X, Y ] when X, Y ∈ B and C (X, Y ) = [X, Y ] when X, Y ∈ C . The category B is pre-additive (enriched over Ab), and C is additive (has biproducts). Let PB and PC denote the categories of abelian presheaves defined on B and C . They consist of the additive functors from B or C op to Ab and the additive natural transformations between them. 1.1. Definition. Define the Freyd functor F : T →PB by sending an object X to the functor FX specified on objects and morphisms of B by FX(−) = [−, X] and sending a We thank Grigory Garkusha for the observation following Corollary 3.3, and we thank an anonymous referee for offering simplifications of the formal arguments leading up to Proposition 2.1. Received by the editors 2011-10-19 and, in revised form, 2012-08-30. Transmitted by Ross Street. Published on 2012-09-06. 2010 Mathematics Subject Classification: 18E30, 55P42.