path groupoid

For $X$ a smooth space, there are useful refinements of the fundamental groupoid $\Pi_1(X)$ which remember more than just the homotopy class of paths, i.e. whose morphisms are (piecewise, say) smooth paths in $X$ modulo an equivalence relation still strong enough to induce a groupoid structure, but weaker than dividing out homotopies relative to endpoints.

Let $X$ be a smooth manifold.

For $\gamma_1, \gamma_2 : [0,1] \to X$ two smooth maps, a **thin homotopy** $\gamma_1 \Rightarrow \gamma_2$ is a smooth homotopy, i.e. a smooth map

$\Sigma : [0,1]^2 \to X$

with

- $\Sigma(0,-) = \gamma_1$
- $\Sigma(1,-) = \gamma_2$
- $\Sigma(-,0) = \gamma_1(0) = \gamma_2(0)$
- $\Sigma(-,1) = \gamma_1(1) = \gamma_2(1)$

which is *thin* in that it doesn’t sweep out any surface: every $2$-form pulled back to it vanishes:

- $\forall B \in \Omega^2(X)\colon \Sigma^* B = 0$.

A path $\gamma\colon [0,1] \to X$ has **sitting instants** if there is a neighbourhood of the boundary of $[0,1]$ such that $\gamma$ is locally constant restricted to that.

The **path groupoid** $P_1(X)$ is the diffeological groupoid that has

- $Obj(P_1(X)) = X$
- $P_1(X)(x,y) = \{$thin-homotopy classes of paths $\gamma\colon x \to y$ with sitting instants$\}$.

Composition of paths comes from concatenation and reparameterization of representatives. The quotient by thin-homotopy ensures that this yields an associative composition with inverses for each path.

This definition makes sense for $X$ any generalized smooth space, in particular for $X$ a sheaf on Diff.

Moreover, $P_1(X)$ is always itself naturally a groupoid internal to generalized smooth spaces: if $X$ is a Chen space or diffeological space then $P_1(X)$ is itself internal to that category. However, even if $X$ is a manifold, $P_1(X)$ will not be a manifold, see smooth structure of the path groupoid for details.

There are various generalizations of the path groupoid to n-groupoids and ∞-groupoids. See

If $G$ is a Lie group, then internal (i.e. smooth) functors from the path groupoid to the one-object Lie groupoid corresponding to $G$ are in bijection to $Lie(G)$-valued differential forms on $X$. With gauge transformations regarded as morphisms between Lie-algebra values differential forms, this extends naturally to an equivalence of categories

$[P_1(X), \mathbf{B}G] \simeq \Omega^2(X, Lie(G))$

where on the left the functor category is the one of internal (smooth) functors.

More generally, smooth anafunctors from $P_1(X)$ to $\mathbf{B}G$ are canonically equivalent to smooth $G$-principal bundles on $X$ with connection:

$Ana(P_1(X), \mathbf{B}G) \simeq G Bund_\nabla(X)
\,.$

See also

Revised on October 7, 2012 17:33:18
by Urs Schreiber
(89.204.137.246)