# nLab path groupoid

### Context

#### Differential geometry

differential geometry

synthetic differential geometry

∞-Lie theory

# Contents

## Idea

For $X$ a smooth space, there are useful refinements of the fundamental groupoid ${\Pi }_{1}\left(X\right)$ 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 ${\gamma }_{1},{\gamma }_{2}:\left[0,1\right]\to X$ two smooth maps, a thin homotopy ${\gamma }_{1}⇒{\gamma }_{2}$ is a smooth homotopy, i.e. a smooth map

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

with

• $\Sigma \left(0,-\right)={\gamma }_{1}$
• $\Sigma \left(1,-\right)={\gamma }_{2}$
• $\Sigma \left(-,0\right)={\gamma }_{1}\left(0\right)={\gamma }_{2}\left(0\right)$
• $\Sigma \left(-,1\right)={\gamma }_{1}\left(1\right)={\gamma }_{2}\left(1\right)$

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

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

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

###### Definition

The path groupoid ${P}_{1}\left(X\right)$ is the diffeological groupoid that has

• $\mathrm{Obj}\left({P}_{1}\left(X\right)\right)=X$
• ${P}_{1}\left(X\right)\left(x,y\right)=\left\{$thin-homotopy classes of paths $\gamma :x\to y$ with sitting instants$\right\}$.

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, ${P}_{1}\left(X\right)$ is always itself naturally a groupoid internal to generalized smooth spaces: if $X$ is a Chen space or diffeological space then ${P}_{1}\left(X\right)$ is itself internal to that category. However, even if $X$ is a manifold, ${P}_{1}\left(X\right)$ 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

## 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 $\mathrm{Lie}\left(G\right)$-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

$\left[{P}_{1}\left(X\right),BG\right]\simeq {\Omega }^{2}\left(X,\mathrm{Lie}\left(G\right)\right)$[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 ${P}_{1}\left(X\right)$ to $BG$ are canonically equivalent to smooth $G$-principal bundles on $X$ with connection:

$\mathrm{Ana}\left({P}_{1}\left(X\right),BG\right)\simeq G{\mathrm{Bund}}_{\nabla }\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$Ana(P_1(X), \mathbf{B}G) \simeq G Bund_\nabla(X) \,.