Some constructions of exact sequences

A. V. Stepanov

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 7, Tome 272 (2000), pp. 303-320

Résumé

The article presents a construction of a 5-term relative exact sequence in pure categorical terms and the Mayer–Vietoris sequence for weak $\infty$-groupoids of R. Street. In the first half it is shown that several well-known exact sequences can be obtained using Z-diagram described at the beginning of the article. There are a number of categories which are close to the category of topological spaces in the sense of homotopy theory. The objects of such a category can be viewed as weak $\infty$-groupoids in a very intuitive way. Therefore, the language of weak $\infty$-groupoids seems to be very convenient to make explicit constructions in categories of this type. The article presents a combinatorial construction of the path space of a weak $\infty$-groupoid and apply the machinery of Section 1 to get the long exact sequence of homotopy fiber. The construction is quite natural and gives us information on the structure of the relative terms. In the last section the Mayer–Vietoris sequence of a fibre square of weak $\infty$-groupoids is obtained under certain conditions. Of course, this construction makes sense for topological spaces or for any category mentioned in the previous paragraph. But the formulation of the conditions is more natural in terms of weak $\infty$-groupoids. This sequence generalizes the sequence of fibration and in this case the conditions given in the article are equivalent to the homotopy lifting property.