Homotopies of nullhomotopies in a module category. I
In the first author's ``On the category of homotopy pairs'' [Topology Appl. 14, 59-69 (1982; Zbl 0499.55002)], the homotopy category of homotopy pairs of topological spaces based on homotopy commutative squares with a fixed track \((=relative\) homotopy class of a homotopy) has been introduced by the first author as a homotopy coherent alternative to the classical homotopy category of pairs in the sense of Eckmann-Hilton [\textit{B. Eckmann} and \textit{P. J. Hilton}, C. R. Acad. Sci., Paris 246, 2444-2447, 2555-2558, 2991-2993 (1958; Zbl 0092.399-401)] based on strictly commutative squares and homotopies preserving this structure. The importance of the new concept (e.g. for the theory of Toda brackets) has been demonstrated in a series of papers [e.g. the first author, Topological topics, Lond. Math. Soc. Lect. Note Ser. 86, 88-102 (1983; Zbl 0544.55015); the first author and \textit{A. V. Jansen}, Lect. Notes Math. 915, 112-126 (1982; Zbl 0485.55010), Quaest. Math. 6, 107-128 (1983; Zbl 0522.55011); the first author, ``Approximating the homotopy sequence of a pair of spaces'', Tsukuba Math. J. (to appear)]. In the present paper the analogue for injective homotopy theory of modules over a ring [see \textit{P. J. Hilton}, Symp. Int. Topología Algebráica, 273-281 (1958; Zbl 0091.041)] is presented. Due to special features of the homotopy theory of modules such as additivity, the development turns out to be simpler than in the topological case. Applications are given to secondary composition operations (Toda brackets) for modules. Applications to constructions such as a Whitehead product, a Hopf construction and a Hopf invariant for modules are deferred to subsequent papers.
scientific article; zbMATH DE number 4199032
scientific article