Homotopical algebra in homotopical categories (Q1338775)

The author has written a first paper on categorical bases of homological and homotopical algebra [Cah. Topologie Géom. Différ. Catégoriques 33, No. 2, 135-175 (1992; Zbl 0814.18006)]. Here is presented a homotopical algebra based on homotopy kernels and cokernels. The framework is an \(h\)-category in which homotopical algebra is established as an enrichment of homological algebra. Notions of \(h\)-, \(h1\)-, \dots, \(h4\)-categories are also introduced through enrichment of the vertical structure of homotopies. The strongest notion is a sort of relaxed 2- category. Homotopy pullbacks and homotopical diagrammatical lemmas are studied in this setting. Then are introduced (right) semi-homotopical categories (\(h\)-categories with terminal object and homotopy cokernel i.e. mapping cones) and (right) homotopical categories (\(h4\)-categories verifying second order regularity properties for \(h\)-cokernels). In this context is studied the Puppe (cofibration) sequence of a map linked to the tower of iterated \(h\)-cokernels and the \(h\)-cogroup structure of the suspension. Left homotopical categories, based on homotopy kernels, give the fibration sequence of a map and the \(h\)-group of loops. The self-dual notion of homotopical categories is considered together with their stability properties.