Geometric realization and its variants
Given a well-pointed topological group \(G\), the classifying space \(BG\) can be constructed via the geometric realization \(|N_\bullet G|\) of the nerve \(N_\bullet G\) of the associated one-object category \(\mathcal{G}\) [\textit{G. Segal}, Publ. Math., Inst. Hautes Étud. Sci. 34, 105--112 (1968; Zbl 0199.26404)]. On the other hand, whether \(G\) is well-pointed or not, Milnor's construction based on the \(n\)-fold join \(G\ast\cdots\ast G\) always yields the homotopy type of \(BG\). Then, by \textit{T. tom Dieck} [Manuscr. Math. 11, 41--49 (1974; Zbl 0283.55006)] those two constructions (under some conditions) are homotopy equivalent. Milnor's construction is generalized to topological groupoids by \textit{A. Haefliger} [Lect. Notes Math. 197, 133--163 (1971; Zbl 0215.52403)] and to categories internal to the category of spaces by Segal [loc. cit.]. The author compares different realization functors for simplicial objects in a topologically enriched model category. Then, the tom T. Dieck's theorem on homotopy type of classifying spaces is generalized in {Theorem 1.1.}. The main ingredient in the proof of that result is a generalization of Segal's lemma [\textit{G. Segal}, Topology 13, 293--312 (1974; Zbl 0284.55016)] for \(\mathcal{T}op\)-model categories.
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