The holonomy groupoid of a foliation

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $F$ be a regular foliation of $M$. Two leafwise paths with the same endpoints $x,y$ are holonomy-equivalent when their holonomy germs relative to some choice of local transversals at $x$ and $y$ agree (The holonomy germ is independent of the foliation chart chain, Local transversals to a regular foliation). By chain independence the answer does not depend on the choice of the transversals: by the concatenation law Holonomy respects path concatenation and reversal, passing from the pair of transversals $(T,T^{\mathrm{\prime }})$ to another pair $(T_1,T_1^{\mathrm{\prime }})$ replaces the germ $h_c(T^{\mathrm{\prime }},T)$ of any leafwise path $c$ from $x$ to $y$ by $\beta \circ h_c(T^{\mathrm{\prime }},T)\circ {\alpha }^{-1}$, where $\alpha$ and $\beta$ are the germs of constant-path transport $\alpha :(T,x)\to (T_1,x)$ and $\beta :(T^{\mathrm{\prime }},y)\to (T_1^{\mathrm{\prime }},y)$ across the plaques at the endpoints; since the same two germs $\alpha ,\beta$ occur for every such path $c$, the relation "the two germs agree" is the same for the two choices. So "one choice" and "every choice" give the same relation.

The holonomy groupoid $\mathrm{Hol}(F)$ is the groupoid with object set $M$ whose arrows from $x$ to $y$ are the holonomy classes of leafwise paths from $x$ to $y$; there is no arrow between points in different leaves. Composition is induced by concatenation of leafwise paths, the identity at $x$ is the class of the constant path, and inverses are induced by reversal. That these operations are well defined on holonomy classes, and that $\mathrm{Hol}(F)$ is a groupoid, is the content of Holonomy classes form a groupoid congruence ↗, which is recorded as the well-definedness certificate of this definition. By Holonomy depends only on leafwise homotopy relative to endpoints and Holonomy respects path concatenation and reversal leafwise homotopy relative to endpoints refines the holonomy relation, and multiplicativity of holonomy germs is what makes composition descend.

Thus $\mathrm{Hol}(F)$ is the quotient of the monodromy groupoid $\mathrm{Mon}(F)$ of The monodromy groupoid of a foliation by the relation that identifies arrows with equal holonomy germs: the projection $\mathrm{Mon}(F)\to \mathrm{Hol}(F)$ sends the leafwise homotopy class of a path to its holonomy class. Arrows whose endpoints are not composable have no composite. As for the monodromy groupoid, no topology on the arrow set is imposed and no smooth structure on it is asserted.