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'.
Keywords: Homotopy theory, abstract homotopy theory, 2-categories, cofibrations, fibre spaces, chain complexes.
2000 MSC: 55U35, 18G55, 18D05, 55P05, 55R05, 55U15.
Theory and Applications of Categories, Vol. 9, 2001, No. 2, pp 17-42.