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'.
Theory and Applications of Categories, Vol. 9, 2001, No. 2, pp 17-42 http://www.tac.mta.ca/tac/volumes/9/n2/n2.dvi http://www.tac.mta.ca/tac/volumes/9/n2/n2.ps http://www.tac.mta.ca/tac/volumes/9/n2/n2.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n2/n2.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n2/n2.psTAC Home