As a geometric interpretation of the simplicial fundamental groupoid, resp. its equivalent relation I thought about something like arcwise differentiable (or arcwise linear) paths modulo "we can go over to others, by gluing triangles (2-simplexes) onto the path". Is this a good geometric interpretation of the simplicial fundamental groupoid?.
Is there any sort of similar interpretation for higher simplicial homotopy group(oids)? It should be something like gluing tetrahedrons together in the right way.
I know this is maybe a too vague question to answer.