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In his book Classifying spaces and classifying topoi Moerdijk proves that there exists a wieak homotopy equivalence between the classifying topos of the Haefliger space $\Gamma^q$ and that of the monoid $M(\mathbb R^q)$ of smooth embeddings of $\mathbb R^q$ into itself.

From this, he concludes that the corresponding classifying spaces $B\Gamma^q$ and $BM(\mathbb R^q)$ are weakly homotopy equivalent, using

Let $C$ be a topological category. Then there is a weak homotopy equivalence $f:Sh(BC)\to\mathcal B C$, where $\mathcal B C$ represents the classifying topos of $C$.

My problem is passing from $Sh(BC)$ to $BC$. In fact, since $BC=|Nerve(C)|$ has a base of contractible open sets (true?) its homotopy groups coincide with the étale homotopy (pro)groups of $Sh(BC)$. But the definition of weak homotopy equivalence requires the existence of a continuous function between topological spaces inducing those isomorphism in homotopy groups. So I would need some $$g:B\Gamma^q\to BM(\mathbb R^q)$$ inducing isomorphisms in homotopy, and I fail to see how to recover this from the abstract functor between the topoi of sheaves $$Sh(B\Gamma^q)\to\mathcal B \Gamma^q\to \mathcal BM(\mathbb R^q)\to Sh(BM(\mathbb R^q)).$$

Can someone give a hint? Thank you in advance.

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I'll follow the terminology in the book: spaces are sober topological spaces, toposes are Grothendieck toposes, and topos morphisms are geometric morphisms.

For spaces $X,Y,$ isomorphism classes of topos morphisms $Sh(Y)\to Sh(X)$ correspond to continuous maps $Y\to X$ - see Section I.2. The end of Section I.4 gives references to Artin-Mazur for the argument that for locally contractible spaces, the topos morphism is a weak homotopy equivalence if and only if the continuous map is a weak homotopy equivalence. The local contractibility of $B\mathbb C$ follows from the paragraph after IV.2.2. (Artin-Mazur Theorem 12.1 actually assumes paracompactness as well; I'm not sure if that's a problem.)

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  • $\begingroup$ Thank you very much. Also, is it true that the classifying spaces are CW-complexes (hence the weak equivalence can be promoted to a strong equivalence by Whitehead's theorem)? $\endgroup$
    – W. Rether
    Jun 22, 2018 at 20:50

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