A note on exactness and stability in homotopical algebra
Theory and Applications of Categories [electronic only] , CT2000, Tome 9 (2000), pp. 17-42.

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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'.
Classification : 55U35, 18G55, 18D05, 55P05, 55R05, 55U15.
Mots-clés : Homotopy theory, abstract homotopy theory, 2-categories, cofibrations, fibre spaces, chain complexes. 
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     author = {Marco Grandis},
     title = {A note on exactness and stability in homotopical algebra},
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Marco Grandis. A note on exactness and stability in homotopical algebra. Theory and Applications of Categories [electronic only]
, CT2000, Tome 9 (2000), pp. 17-42. https://geodesic-test.mathdoc.fr/item/TAC_2000__9_a1/