Coherent Functors in Stable Homotopy Theory

such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [1], but it is also implicit in the work of Freyd on stable homotopy [9]. In this paper we characterize coherent functors in a number of ways and use them to study a wider class of functors S → Ab which share a weak exactness property. Another purpose of this paper is to investigate certain subcategories of S which are defined in terms of coherent functors. In the category ModΛ of modules over an associative ring Λ, the analogue of a compact object is a finitely presented module. This fact can be made precise (cf. the Appendix), and one has in this context the following classical result: a functor F : ModΛ → Ab is coherent precisely if F preserves products and filtered colimits. There is no obvious way to formulate such a characterization for compactly generated triangulated categories because filtered colimits rarely exist in triangulated categories. Nevertheless, we are able to characterize the coherent functors as follows.