Classifying Spaces of Topological Monoids and Categories
Copyright policy )Classifying Spaces of Topological Monoids and Categories
This article explores homology and weak homotopy equivalences between classifying spaces of topological categories and discrete categories. Many recent results have dealt with this phenomenon: e.g. for any space X there is a discrete group and a homology equivalence BG\(\to X\) [\textit{D. Kan} and \textit{W. Thurston}, Topology 15, 253-258 (1976; Zbl 0355.55004)]; also there is a discrete monoid M and a weak homotopy equivalence BM\(\to X\) [\textit{D. McDuff}, ibid. 18, 313-320 (1979; Zbl 0429.55009)]. The author presents a ''unified'' approach to the theory of comparing categories to their ''discretizations'', culminating in an axiomatic characterization of classifying space functors - on topological monoids they are completely determined by their restrictions to discrete free monoids. Along the way the author proves the McDuff result, recovers \textit{F. Waldhausen}'s characterization of algebraic K-theory, [Ann. Math., II. Ser. 108, 135-204 (1978; Zbl 0397.18012)], and gives a simple proof that the classifying space of the James construction on a space is homotopy equivalent to its suspension. - The article is well written and organized and provides the interested reader with a good way to get acquainted with recent results in classifying spaces.
classifying space of the James construction, Simplicial sets and complexes in algebraic topology, topological monoids, Classifying spaces of groups and \(H\)-spaces in algebraic topology, Categories of topological spaces and continuous mappings, classifying spaces of topological categories, weak homotopy equivalences, discrete monoid, Topological categories, foundations of homotopy theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), algebraic K- theory, discrete categories
classifying space of the James construction, Simplicial sets and complexes in algebraic topology, topological monoids, Classifying spaces of groups and \(H\)-spaces in algebraic topology, Categories of topological spaces and continuous mappings, classifying spaces of topological categories, weak homotopy equivalences, discrete monoid, Topological categories, foundations of homotopy theory, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), algebraic K- theory, discrete categories
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