nLab
path infinity-groupoid

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

• path groupoids

• path n-groupoids.

Definition

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

$${\Delta }_{\mathrm{SmoothSp}}:\Delta \to \mathrm{SmoothSp}$$\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 ${\Delta }^n\subset {ℝ}^n$.

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

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

with values in smooth ∞-stacks given by

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

$$N(X):U\mapsto {\mathrm{Ex}}^{\mathrm{\infty }}\mathrm{SmoothSp}(U\times {\Delta }_{\mathrm{SmoothSp}}^{•},X),$$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 $\mathrm{\infty }$-groupoid is here.

References

A more detailed account is at

• path ∞-groupoid