The Dwyer–Kan loop groupoid (Dwyer-Kan 84) of a simplicial set is a simplically enriched groupoid whose objects are the vertices of and the simplicial set of paths between two such picks up the composable ‘strings’ of higher dimensional simplices where the zeroth vertex is thought of as the domain vertex and the first vertex as the codomain.
This construction establishes an equivalence between the homotopy theory of simplicial groupoid and the classical homotopy theory of simplicial sets (exhibiting both as models for infinity-groupoids). It generalizes the simplicial loop space functor from reduced simplicial sets to simplicial groups.
The loop groupoid functor of Dwyer and Kan is a functor
where the two functions, , source, and , target, are and with relations for .
The face and degeneracy maps are given on generators by
, for , , and
This simplicial groupoid is a simplicially enriched groupoid, as the face and degeneracy operators are constant on the objects.
The original reference is
but beware, there are some typographic errors in key formulas.