2000/08/31 by Grandis, Marco
#18G55 #55U15 #55U35 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0008236
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'.