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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 $H(BGamma)$ of the Haefliger classifying space of $Gamma$, then the continuous classifying map $c:M\to BGamma$ induces an equivalence. Precisely, $H(BGamma)$ 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 $H(BGamma)$.

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 $H(B\Gamma)$.

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?

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 $H(BGamma)$ of the Haefliger classifying space of $Gamma$, then the continuous classifying map $c:M\to BGamma$ induces an equivalence. Precisely, $H(BGamma)$ 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 $H(BGamma)$.

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(BGamma_q). Let c:M->BGamma_q be the Haefliger classifying map of F. Then, F is the pullback of the universal foliation on BGamma_q (whatever this means) through c. In particular, there is an induced groupoid morphism C: H(F)->H(BGamma_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?

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?