Fibred sites and stack cohomology (Q2457752)

Let \(G\) be a stack defined on a Grothendieck site \({\mathcal C}\). Recall that \(G\) is a pseudo-functor on \({\mathcal C}\), with values in groupoids, such that the objects of \(G\) form a pseudo-functorial sheaf. In this general setting, there is a fundamental construction of another Grothendieck site \({\mathcal C}/G\), which is called the site associated to the stack \(G\). The introduction of the site \({\mathcal C}/G\), the idea of which goes back to A. Grothendieck himself, allows to develop the notion of stack cohomology of \(G\) with coefficients in a sheaf \(A\) on \({\mathcal C}/G\) [cf.: \textit{G. Laumon} and \textit{L. Moret-Bailly} [``Algebraic spaces'', Ergebn. Math. Grenzgeb., 3. Folge, Vol. 39 (2000; Zbl 0945.14005)], and this kind of (generalized) cohomology theory is closely related to the categorical homotopy theory of sheaves and stacks via the concept of Eilenberg-MacLane simplicial sheaves \(K(A,n)\) associated to a sheaf \(A\) on \({\mathcal C}/G\) [cf.: \textit{J. F. Jardine}, Homology Homotopy Appl. 3, No. 2, 361--384 (2001; Zbl 0995.18006)]. In the paper under review, the author provides the technical fundamentals for a conceptual expansion of this (already quite general) framework. In the first part, it is shown that the usual construction of the site \({\mathcal C}/G\) associated to a stack \(G\) admits a substantial generalization, namely to a site \({\mathcal C}/ A\) fibred over a presheaf of categories \(A\) on the base site \({\mathcal C}\). This concept generalizes many standard constructions, including those of the geometric sites fibred over diagrams of algebraic schemes. Then the simplicial presheaves on such a site \({\mathcal C}/A\) are described as enriched contravariant diagrams on \(A\). From this characterization it is derived that, if \(A\) is a presheaf of groupoids \(G\), then the associated homotopy category of simplicial presheaves over \({\mathcal C}/G\) is Quillen equivalent to the homotopy category of simplicial presheaves over the nerve \(\text{BG}^{\text{op}}\) of the opposite presheaf of categories \(G^{\text{op}}\). This implies that the homotopy theory for the ``fibred site'' \({\mathcal C}/ G\) is an invariant of the homotopy type of the presheaf \(G\), and the following number of related homotopy invariance results are basically obtained by applying suitable strings of classical Quillen equivalences with respect to particular other types of presheaves. An additional consequence of the author's general approach is that stack cohomology (in the sense mentioned above) can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type. This is advantageous in view of the fact that such a presheaf \(G\) is often more concretely defined than its associated stack, and that the fibred site \({\mathcal C}/G\) appears to be more explicit than the related Grothendieck site \({\mathcal C}/\text{Stack}(G)\). As the author points out, the results of the present paper should be seen as the starting point for further studies on homotopy invariance properties of this kind, for example with a view toward categories of chain complexes and other more general objects.