A Haefliger structure on a smooth manifold is a cocycle with coefficient in the Haefliger groupoid $\Gamma^q$. This generalises the notion of foliation of codimension $q$. We know that the classifying space of $\Gamma^q$ has a classifying property with respect to $\Gamma^q$-structures (Haefliger's theorem). From the other side, we know that by Diaconescu's Theorem (cfr. Moerdijk, Classifying spaces and classifying topoi) the classifying topos $\mathcal B\Gamma^q$ of $\Gamma^q $ "classifies" principal $\Gamma^q$-bundles, in the sense that for a topological space $X$ one has $$Hom(Sh(X),\mathcal B\Gamma^q)\cong Prin(X, \Gamma^q).$$ In fact, Moerdijk shows that $B\Gamma^q$ and $\mathcal B\Gamma^q$ are weakly homotopy equivalent. This leads me to the question: if $X$ is a smooth manifold, can we say that $Prin(X,\Gamma^q)$ and the category of Haefliger structures are related (equivalent, weakly homotopy equivalent)? Note that this won't probably arise from the preceding facts, since the "classifying property" of $B\Gamma^q$ is more complicated than that of $\mathcal\Gamma^q$. I am asking only by suggestion of the situation.

This could actually be understood as a generalisation of this question, passing from Lie groups to topological groupoids (namely $\Gamma^q$).

Thank you in advance.


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