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A note on exactness and stability in homotopical algebra

##
Marco Grandis

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.

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