What is the relation between the holonomy groupoid of a foliation and the corresponding Haefliger groupoid? Given a foliation, there is a holonomy groupoid and a classifying map
to the Haefliger classifying space via the Haefliger groupid. What is the relation between these groupids?
 A: Write $F$ your foliation, $M$ its ambiant manifold, $q=dim(M)-dim(F)$ its codimension. The holonomy groupoid $H(F)$, if I'm correct, is the set of classes of triples $(x,\gamma,y)$ where $\gamma$ is a tangential path; and $(x,\gamma,y)~(x,\gamma',y)$ iff $\gamma$ and $\gamma'$ have the same holonomy.
I'm not expert enough in groupoids to use the proper vocabulary, but the relation is as follows between the holonomy groupoid $H(F)$ of your foliation and the universal groupoid $H(B\Gamma_q)$. Let $c:M\to B\Gamma_q$ be the Haefliger classifying map of $F$. Then, $F$ is the pullback of the universal foliation on $B\Gamma_q$ (whatever this means) through $c$. In particular, there is an induced groupoid morphism $C: H(F)\to H(B\Gamma_q)$. Moreover, one has a partial injectivity property: for $(x,\alpha,y)$ and $(x,\beta,y)$ in $H(F)$ with the same endpoints, one has $C(x,\alpha,y)=C(x,\beta,y)$ iff $\alpha=\beta$.
Does this help?
On the other hand, if you rather mean to compare the holonomy groupoid $\Gamma=H(F)$ of the given foliation with the holonomy groupoid $H(B\Gamma)$ of the Haefliger classifying space of $\Gamma$, then the continuous classifying map $c:M\to B\Gamma$ induces an equivalence. Precisely, $H(B\Gamma)$ is simple (at most 1 arrow between two units) and $c$ induces a bijection between the set of orbits of $\Gamma$ and the set of orbits of $H(B\Gamma)$.
