On the cohomology of categories, universal Toda brackets and homotopy pairs
For a category \({\mathcal C}\) a natural transformation of cohomology groups \[ \lambda: H^{n+1} ({\mathcal C}, D) \to H^n(Pair({\mathcal C}), D^\#) \] is defined. Here \(Pair({\mathcal C})\) is the category of pairs in \({\mathcal C}\), \(D\) is a natural system on \({\mathcal C}\), i.e., a functor from the category of factorizations of \({\mathcal C}\) to the category of abelian groups, and \(D^\#\) is the natural system induced on the category of pairs. If \({\mathcal C}\) is a homotopy category of suspensions an element, called universal Toda bracket, \(\langle{\mathcal C} \rangle_\Sigma \in H^3 ({\mathcal C}, D_\Sigma)\) is defined which determines all classical Toda brackets on \({\mathcal C}\) [cf. \textit{H.-J. Baues} and \textit{W. Dreckmann}, ``The cohomology of homotopy categories and the general linear group'', \(K\)-Theory 3, No. 4, 307-338 (1989; Zbl 0701.18009)]. Here the natural system \(D_\Sigma\) is defined on objects by \(D_\Sigma (h)= [\Sigma A,B]\) if \(h:A \to B\) is a map in \({\mathcal C}\). As a central result it is shown that \(\lambda \langle {\mathcal C} \rangle_\Sigma \in H^2 (Pair({\mathcal C}),\;D^\#_\Sigma)\) characterizes the linear extension \(D^\#_\Sigma \to Hopair({\mathcal C}) \to Pair ({\mathcal C})\) where \(Hopair({\mathcal C})\) is the category of homotopy pairs in the sense of Hardie [\textit{K. A. Hardie}, ``On the category of homotopy pairs'', Topology Appl. 14, 59-69 (1982; Zbl 0499.55002)]. Explicit computations of universal Toda brackets are included.
scientific article; zbMATH DE number 1036284