  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'.
