The path ∞-groupoid of a generalized smooth space is a smooth version of the fundamental ∞-groupoid of . Its truncations to lower categorical degree yield
One way to define a path ∞-groupoid in terms of Kan complexes is to let
be the canonical cosimplicial object in smooth spaces that sends the abstract -simplex to the standard smooth -simplex .
As every cosimplicial object with values in a category with colimits this induces a notion of nerve and realization. The smooth nerve operation
with values in smooth ∞-stacks given by
where on the right we have a simplicial object in the category of smooth spaces regarded as a model for a smooth ∞-stack.
Notice that the Kan complex valued sheaf presented by this is given for instance by the simplicial sheaf
which can be thought of as having in degree piecewise smooth -dimensional paths.
Functors out of the path groupoid and path n-groupoid represent connections and higher connectios. Discussion of this for the path -groupoid is here.
A more detailed account is at
Revised on December 21, 2009 14:00:01
by Urs Schreiber