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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
##### Keywords:
Kan loop group; ordered graph; nerve; simplicial group
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