6.4 Multi-simplicial spaces
Let r\geqslant0 be an integer. An r-simplicial object Z_\bullet in a category \mathscr{C} is a contravariant functor from the r-fold product (\Delta)^r to \mathscr{C}. The diagonal simplicial object \delta Z_\bullet is the composite functor (\Delta)\to(\Delta)^r\to\mathscr{C}.
As in (5.1.6), we define the topos of sheaves on an r-simplicial topological space. Let \Gamma^\bullet be the functor F \mapsto (\Gamma(X_{n_1\ldots n_r},F^{n_1\ldots n_r})) from sheaves on X_\bullet to r-cosimplicial sets. For small r, we often prefer to write \Gamma^{\bullet\ldots\bullet} (with r-many copies of \bullet). We have a co-augmentation \Gamma(X_\bullet,F^\bullet)\to\Gamma^\bullet(X_\bullet,F^\bullet). The functors of cohomology with values in an abelian sheaf F are the derived functors of the “global sections” functor \Gamma. They can be calculated by a procedure parallel to that of (5.2.3), i.e. \mathrm{R}\Gamma =\mathrm{s}\mathrm{R}\Gamma^\bullet \colon \mathrm{D}^+(X_\bullet) \to\mathrm{D}^+((\mathsf{Ab})). \tag{6.4.2.1}
A sheaf F on an r-simplicial topological space X_\bullet induces a sheaf \delta F on the diagonal simplicial space \delta X_\bullet. It follows from the Cartier–Eilenberg–Zilber theorem that \mathrm{R}\Gamma(X_\bullet,K) \xrightarrow{\sim}\mathrm{R}\Gamma(\delta X_\bullet,\delta K). \tag{6.4.2.2}
We restrict to the case r=2. A bisimplicial object 1-augmented over a simplicial object S_\bullet is a contravariant functor from (\Delta^+)\times(\Delta) to \mathscr{C} such that S_\bullet is the composite functor (\Delta) \xrightarrow{(\Delta_{-1},\bullet)} (\Delta^+)\times(\Delta) \to \mathscr{C}. To denote a bisimplicial object 1-augmented over S_\bullet, with underlying bisimplicial object X_{\bullet\bullet}, we will use notation of the form a\colon X_{\bullet\bullet} \to S_\bullet. For n\geqslant0, a\colon(X_{\bullet n})\to S_n is a simplicial object augmented over S_n. If F is a sheaf on X_{\bullet\bullet}, then the (a_*(F|X_{\bullet n})\text{ on }S_n)_{n\geqslant0} form a sheaf on S_\bullet; we thus define a topos morphism a \colon X_{\bullet\bullet}^\sim \to S_\bullet^\sim.
We explain in (Saint-Donat 1972) that \mathrm{R}a_* can be calculated “component by component”, i.e. \mathrm{R}a_*K|S_n = \mathrm{R}a_*(K|X_{\bullet n}). It thus follows that if, for each n, the morphism a\colon(X_{\bullet n})\to S_n is of cohomological descent, then a\colon X_{\bullet\bullet}\to S_\bullet is of cohomological descent: for every complex K\in\mathrm{D}^+(S_\bullet) of abelian sheaves, we have K \xrightarrow{\approx} \mathrm{R}a_*a^*K.
In (Saint-Donat 1972), we show that, for every separated simplicial scheme S_\bullet, there exists a bisimplicial scheme X_{\bullet\bullet} that is 1-augmented over S_\bullet, along with i\colon X_{\bullet\bullet}\hookrightarrow\overline{X}_{\bullet\bullet} such that:
- The \overline{X}_{nm} are smooth and projective; X_{nm} is the complement of a normal crossing divisor D_{nm} in \overline{X}_{nm}, and we can suppose the D_{nm} to be a union of smooth divisors.
- For n\geqslant0, X_{\bullet n} is a proper hypercover of S_n, and we can take it to be s-split.
The construction proceeds as in §6.2, but the induction is more complicated. The claims of uniqueness (6.2.8) remain true, mutatis mutandis.