nLab
path groupoid

Idea

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.

Definition

Let $X$ be a smooth manifold.

Definition

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

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

• $\forall B\in {\Omega }^2(X):{\Sigma }^{*}B=0$.

Definition

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

Definition

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

• $\mathrm{Obj}(P_1(X))=X$
• $P_1(X)(x,y)=\{$thin-homotopy classes of paths $\gamma :x\to y$ 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

• path n-groupoid,

• path ∞-groupoid.

Remarks

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 $\mathrm{Lie}(G)$-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

$$[P_1(X),BG]\simeq {\Omega }^2(X,\mathrm{Lie}(G))$$[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 $BG$ are canonically equivalent to smooth $G$-principal bundles on $X$ with connection:

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

See also

• Atiyah Lie groupoid.