# nLab path groupoid

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

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

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:

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

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.

###### Definition

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

## 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 $Lie(G)$-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:

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