The topological realization of a simplicial presheaf

While preparing for EUROPROJ’s Trento school on stacks (September 1996) it became apparent that an obvious thing that one would like to say about a stack—to take its topological realization—was not altogether obvious to define or to handle. The same question can be posed more generally for a presheaf of spaces or (as it is common to say) a simplicial presheaf. We give a definition of the topological realization of a presheaf of spaces on a site X , with respect to a covariant functor F : X → Top. It is a topological space defined in a fairly obvious way. In our usual case X will a site of schemes over Spec(C) and F is the functor which to a scheme X associates its underlying usual topological space X. If G is representable by an object X then the realization of G is (homotopic to) X. One recovers the topological realization of a simplicial presheaf by first realizing over each object into a presheaf of spaces, and then applying our definition. One recovers the topological realization of a stack (of groupoids) by first strictifying into a presheaf of groupoids then taking the realization of the nerve of the groupoid over each object to get a presheaf of spaces. Once the theory of n-stacks is well off the ground, the same remark will hold for the topological realization of an n-stack (whereas for now one must replace the notion of nstack by n-truncated presheaf of spaces, and from this point of view one can directly take the topological realization as defined below). The main theorem is the invariance of topological realization under Illusie weak equivalence. Recall that if G is a presheaf of spaces on the site X then we obtain a presheaf of sets π 0 (G) on X and for any g ∈ G(X) a presheaf of groups π pre i (G|X/X , g) on X /X. Then π0(G) (resp. πi(G|X/X , g)) is defined to be the sheafification of π pre 0 (G) (resp. the sheafification of π i (G|X/X , g) on X /X). A morphism ψ : G → G ′ of presheaves of spaces is an Illusie weak equivalence if it induces an isomorphism on the homotopy sheaves