Homotopies of nullhomotopies in a module category. I
type: Item
instance of: scholarly article
title: Homotopies of nullhomotopies in a module category. I
zbMATH Open document ID: 0726.55009
DOI: 10.1016/0022-4049(90)90140-D
published in: Journal of Pure and Applied Algebra
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 13C99
Mathematics Subject Classification ID: 18G05
zbMATH DE Number: 4199032
zbMATH Keywords: homotopy category
zbMATH Keywords: Toda brackets
zbMATH Keywords: injective homotopy theory
zbMATH Keywords: modules over a ring
zbMATH Keywords: secondary composition operations
author: K. A. Hardie
author: Peter Hilton
MaRDI profile type: Publication
cites work: Q3268806
cites work: On the category of homotopy pairs
cites work: Q3333734
cites work: Q3945243
cites work: TODA BRACKETS AND THE CATEGORY OF HOMOTOPY PAIRS
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cites work: A Note on Homotopy Equivalences
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description: scientific article; zbMATH DE number 4199032