Homological algebra and (semi)stable homotopy

Abstract

If A is a complete and cocomplete abelian category, which we allow ourselves to conflate with the corresponding representable homotopy theory then the 2-functors HochA, taking the small category C to the homotopy category of chain complexes over A C and Hoch + A, with value the homotopy category of positive chain complexes, are both homotopy theories (in the sense of my monograph, A.M.S. Memoirs 383), the former being stable in the sense that the suspension hyperfunctor is an equivalence, while the latter is semistable. The hyperfunctors res: A → HochA and res +: A → Hoch + A which take an X in A C to a chain complex concentrated in degree 0 may be characterized as “resolvent”. Then the two chain-complex theories associated to A are, respectively, the universal resolvent stabilization and semistabilization of A. In other words, a “universal problem” of stabilization leads, for abelian categories, to the construction of chain complexes, just as a corresponding problem for topological spaces leads to the construction of spectra.