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Homology fibrations and “group-completion” revisited. (English) Zbl 1064.18012
The authors give a proof of the following well known result:
Theorem: Let \(\mathcal{M}\) be a simplicial category and \(F: \mathcal{M}^{op} \to {\mathcal S}paces\) a contravariant diagram. Assume that any morphism \(f: i \to j\) induces an isomorphism in integral homology \(H_{\ast}(F(j);\mathbb{Z}) \to H_{\ast}(F(i);\mathbb{Z})\). Then, for each object \(i \in \mathcal{M}\), the map \(F(i) \to \text{Fib}_{i}(\pi_{\mathcal{M}})\) to the homotopy fibre of the canonical map \(\pi_{\mathcal{M}}: E_{\mathcal{M}}F \to B\mathcal{M}\) over \(i\) is a homology equivalence.
Here, \(E_{\mathcal{M}}F\) is the homotopy colimit for the functor \(F\). One could argue that the assumption that \(F\) is contravariant is not necessary.
The proof given in this paper is advertised to involve minimal use of the homotopy theory of simplicial sets, but the ideas appearing in the proof are standard.