--- algebraic_geometry/src/desirables.tex	2005-07-19 16:11:15.000000000 -0400
+++ submissions/desirables-0001.tex	2005-07-21 13:01:24.000000000 -0400
@@ -93,22 +93,35 @@
 functors that compose on the nose). So in other words functors from the base
 category to the category of groupoids.
 
-\subsection{Sites}
+\subsection{Sites and Topoi}
 \label{subsection-sites}
 
 \noindent
-Do a little bit of theory here. Talk about sheaves, morphisms of sites. 
+Do a little bit of theory here. Talk about sheaves, morphisms of sites.
 The category of sheaves on a site now means all sheaves with values in
 $\text{Sets}_\alpha$ where $\alpha$ is suitably large (relative to the
 site).
 
+Introduce the notion of topos and morphism of topoi. The notion of 
+simplicial and strictly simplicial topos.
+
 \smallskip\noindent
-Ringed sites, quasi-coherent sheaves of modules.
+Ringed sites, quasi-coherent sheaves of modules. Ringed topos and 
+morphism of ringed topoi.
 
 \smallskip\noindent
 Some generalities about cohomology goes in here as well. (Just with
 injective resolutions, nothing fancy. Allthough it might be nice to
-have hypercoverings here for later use.)
+have hypercoverings here for later use.) Injective resolutions of a
+sheaf in a (strictly) simplicial topos.  The fundamental spectral 
+sequence relating cohomology of the individual pieces to global
+cohomology. The simplicial topos arising from a covering of the final
+object in a topos and comparison of cohomologies.
+
+\smallskip\noindent
+Some basic facts about cohomological descent (at least enough to deal
+with a flat hypercover for quasi--coherent sheaves and a proper
+hypercover for \'etale sheaves).
 
 \subsection{Stacks}
 \label{stacks}