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On the construction of the Kan loop group. (English) Zbl 0874.55008
In [Ann. Math., II. Ser. 67, 282-312 (1958; Zbl 0091.36901)] D. M. Kan defined a functor \(G:\) simplicial sets \(\rightarrow\) simplicial groups such that \(G(X)\) has the same homotopy type as the loop space of \(X\), for each simplicial set. The present author provides an alternative description of the simplicial groups \(G(X)\). He points out that the groups \(G_n(X)\) can be taken as fundamental groups of the nerves of suitably defined ordered graphs. For this an “ordered graph” should be viewed as a small category in which two non-identity morphisms are never composable; thus its nerve is a 1-dimensional simplicial set.

MSC:
55P35 Loop spaces
55Q05 Homotopy groups, general; sets of homotopy classes
55U10 Simplicial sets and complexes in algebraic topology
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