On the theory of path groupoids

Functorial path groupoids P(X) are constructed for each simplicial set X generalizing the loop groups G(X) defined by Kan for each reduced simplicial set. Locally transitive simplicial groupoids and adequate refelexive simplicial graphs are introduced in order to prove that the free simplicial groupoid on an adequate graph is a path groupoid. A calculus of extension and restriction for simplicial groupoids and graphs is developed to construct path groupoids from covers of a simplicial set.