# Where does the ''hyper'' go?

There is a model structure on the category $Grpd$ of groupoids such that weak equivalences are equivalences of categories, cofibrations are the injections on objects (see nlab) and there is a Quillen adjunction $$sSets\leftrightarrows Grpd$$ with right adjoint the nerve $N:Grpd\to sSets$ and with the standard model structure on $sSets$.

Let $C$ be a Grothendieck site with enough points and with a Grothendieck topology given by a basis of covers. Let $D$ bei either $sSets$ or $Grpd$ and let $P(C, D)$ denote the category of presheaves on $C$ with values in $D$.

In Hollander's ''A homotopy theory for stacks'', it is shown that there is a model structure on $P(C,Grpd)$ with weak equivalences the stalkwise equivalences of groupoids and fibrant objects precisely the stacks. If I understand correctly, it is obtained by first taking the projective model structure (i.e. weak equivalences and fibrations are defined objectwiese) on $P(C,Grpd)$ and then Bousfield-localizing at those $F$ for which $$F(X)\to\operatorname{holim}F(U_\bullet)$$ is a weak equivalence for all Cech nerves $U_\bullet\to X$ of a cover $\{U_j\to X\}$.

It is further shown in Theorem 5.4. of Hollander's ''A homotopy theory for stacks'' (please correct me, if I am wrong!), that there exists a Quillen adjunction $$A:P(C,sSets)\leftrightarrows P(C,Grpd):B$$ (with right adjoint $B$) induced by the adjunction between simplicial sets and groupoids where $P(C,Grpd)$ has the model structure from above and $P(C,sSets)$ has the local injective model structure of Jardine.

It would be understandable if the left hand side $P(C,sSets)$ of this adjunction would carry the local projective model structure of Blander (''Local model structures on simplicial presheaves''). What is the reason that this (stronger) statement involving the local injective model structure on $P(C,sSets)$ holds?

Dugger, Hollander and Isaksen show in ''Hypercovers and simplicial presheaves'', that the local injective model structure of Jardine on $P(C,sSets)$ is obtained by first taking the injective model structure (i.e. weak equivalences and cofibrations are defined objectwiese) on $P(C,sSets)$ and then Bousfield-localizing at those $F$ for which $$F(X)\to\operatorname{holim}F(U_\bullet)$$ is a weak equivalence for all hypercovers $U_\bullet\to X$.

A Cech nerve $U_\bullet\to X$ of a cover $\{U_j\to X\}$ defines a hypercover but not all hypercovers are of this form. Now $B$ preserves fibrant objects as a right adjoint of a Quillen adjunction. This means that taking a stack $F:C^{op}\to Grpd$ and composing with the nerve $N:Grpd\to sSets$ yields a fibrant simplicial presheaf in the local injective model structure on $P(C,sSets)$.

Why does it suffice to test fibrancy of elements of $P(C,Grpd)$ only on covers and not on hypercovers? Where does the ''hyper'' go?

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Concerning your first question, the problem is showing that $A$ preserves cofibrations, isn't it? I don't see it either, the explanation offered in 'A homotopy theory for stacks' is nor clear enough to me. About your second question, isn't this Corollary A.9 in 'Hypercovers and simplicial presheaves'? In this case, the argument in the proof does look convincing. – Fernando Muro Jun 28 '13 at 17:31
Dear @FernandoMuro, thanks for the comment. The nerve of a groupoid has all $\pi_n$ with $n\geq 2$ trivial, correct? I forgot this, sorry, then the Corollary answers the second question. Do you know if the nerve of a groupoid is $2$-skeletal, i.e. the same as its $2$-skeleton? – Ronald Bernard Jun 28 '13 at 18:26
Yes, nerves of groupoids don't have higher homotopy groups. They are coskeletal rather than skeletal. – Fernando Muro Jun 28 '13 at 18:43
The nerve of any category is $2$-coskeletal, ok. Thank you! – Ronald Bernard Jun 28 '13 at 18:55
The nerve of any groupoid. – Fernando Muro Jun 28 '13 at 19:07