A Note on Exactness and Stability in Homotopical Algebra

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of “homotopical exactness”, appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general “categories with homotopies” having homotopy kernels and cokernels, but become more interesting under suitable “stability” hypotheses, satisfied in particular by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of “homotopical homology”. Introduction The purpose of this work is to investigate the notion of “homotopically exact” sequence in categories equipped with homotopies, pursuing a project of developing homotopical algebra as an enriched version of homological algebra [6, 7]. Well known instances of such sequences are: (a) the cofibre sequence of a map f : A → B, or Puppe sequence [18], for topological spaces or pointed spaces A → B → Cf → ΣA → ΣB → ΣCf → . . . (1) (Cf is the h-cokernel of f , or standard homotopy cokernel, or mapping cone; Σ denotes the suspension) where every map is, up to homotopy equivalence, an h-cokernel of the preceding one; (b) the fibre sequence of a map of pointed spaces, which has a dual construction and properties; (c) the fibre-cofibre sequence of a map f : A → B of chain complexes (Kf is the h-kernel) . . . → ΩA → ΩB → Kf → A → B → Cf → ΣA → ΣB → . . . (2) where both the aforementioned exactness conditions are satisfied, and each three-term part is homotopy equivalent to a componentwise-split short exact sequence of complexes. The drastic simplification of exactness properties in the last example is a product of the Work supported by MIUR Research Projects Received by the editors 2000 August 29 and, in revised form, 2001 November 28. Published on December 5 in the volume of articles from CT2000. 2000 Mathematics Subject Classification: 55U35, 18G55, 18D05, 55P05, 55R05, 55U15.