# Contents

## Idea

The path ∞-groupoid $\Pi(X)$ of a generalized smooth space $X$ is a smooth version of the fundamental ∞-groupoid of $X$. Its truncations to lower categorical degree yield

## Definition

One way to define a path ∞-groupoid in terms of Kan complexes is to let

$\Delta_{SmoothSp} : \Delta \to SmoothSp$

be the canonical cosimplicial object in smooth spaces that sends the abstract $n$-simplex $\Delta[n]$ to the standard smooth $n$-simplex $\Delta^n \subset \mathbb{R}^n$.

As every cosimplicial object with values in a category with colimits this induces a notion of nerve and realization. The smooth nerve operation

$N : SmoothSp \to SmoothSp^{\Delta^{op}}$

with values in smooth ∞-stacks given by

$N(X) : U \mapsto SmoothSp(U \times \Delta^\bullet_{SmoothSp}, X) \,,$

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

$N(X) : U \mapsto Ex^\infty SmoothSp(U \times \Delta^\bullet_{SmoothSp}, X) \,,$

which can be thought of as having in degree $k$ piecewise smooth $k$-dimensional paths.

## Connections

Functors out of the path groupoid and path n-groupoid represent connections and higher connectios. Discussion of this for the path $\infty$-groupoid is here.

## References

A more detailed account is at

Revised on December 21, 2009 14:00:01 by Urs Schreiber (80.187.144.92)