Topological representation of sheaf cohomology of sites, by Carsten Butz and Ieke Moerdijk

For a site S (with enough points), we construct a topological space X_S and a full embedding \varphi^* of the category of sheaves on S into those on X_S (i.e., a morphism of toposes \varphi : Sh(X_S) --> Sh(S) ). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms

	 *	   *	   *
	H (S,A) = H (X ,phi A)
for any abelian sheaf A on S. As a particular case, this will give for any scheme Y a topological space X_(Y) and a functorial isomorphism between the \'etale cohomology H*(Y_{et},A) and the ordinary sheaf cohomology H^*(X_(Y),\varphi^*A), for any sheaf A for the \'etale topology on Y.


Carsten Butz <>
Ieke Moerdijk <>