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On the geometric bar construction and the Brown representability theorem. (English) Zbl 0926.55011
This is one of a series of papers following on from P. I. Booth [Equivalent homotopy theories and groups of self-equivalences, Lect. Notes Math. 1425, 1-16 (1990, Zbl 0706.55004)]. The author’s overall objective is to develop a theory of enriched fibrations and their classifying spaces that combines and incorporates many individual theories, including the “classical” one of principal, Hurewicz and sectioned fibrations.
In this paper the author relates universal enriched quasi-fibrations – produced using the two-sided geometric bar construction of J. May – to universal enriched fibrations obtained via the Brown Representability Theorem in the author’s previous papers. These complementary methods of producing classifying spaces are blended together, thereby producing a classification theory with the advantages of both approaches and the disadvantages of neither.
More precisely, let \(\mathcal E\) be a category of enriched topological spaces. Suppose \(F\) is a given \(\mathcal E\)-space and denote by \(\mathcal F\) the category of fibres containing \(F\). J. May used a \(\Gamma\)-completeness assumption to convert a universal \(\mathcal F\)-quasi-fibration \(q_{\mathcal F}: Y_{\mathcal F}\to C_{\mathcal F}\) (constructed by means of the bar construction) into a universal \(\mathcal F\)-fibration. The \(\Gamma\)-completeness assumption, however, has the drawback that its use detracts from the simplicity and generality of the classification result obtained. The Brown Representability Theorem approach allows the construction of an alternative universal \(\mathcal F\)-fibration \(p_{\mathcal F}: X_{\mathcal F}\to B_{\mathcal F}\). This procedure avoids the above problems, but \(p_{\mathcal F}\) cannot be applied in the same direct and flexible fashion that is possible for the former approach. The author uses a fibred mapping space argument to equate \(p_{\mathcal F}\) and \(q_{\mathcal F}\). Namely, he constructs weak homotopy equivalences \(h:X_{\mathcal F}\to Y_{\mathcal F}\) and \(g:B_{\mathcal F}\to C_{\mathcal F}\) such that \(gp_{\mathcal F}=q_{\mathcal F}h\).
Reviewer: T.E.Panov (Moskva)

MSC:
55R65 Generalizations of fiber spaces and bundles in algebraic topology
55R05 Fiber spaces in algebraic topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
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References:
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