There is an equivalent to the espace etale construction for Grothendieck toposes, which only works (unsurprisingly) if the topos in question has enough points. It is a well-kept secret in category theory, mentioned briefly only here and there (a single line in Johnstone's textbook, also van Osdol's article in SLN 235 (I think this is right), the book titled "exact categories and categories of functors", Barr and van Osdol eds.) The idea is that if a topos E defined over a base topos S has enough points (and here we can take E to be an ordinary Grothendieck topos and S the category of sets) then E is comonadic over a certain S/I, I being an object of S. In other words, and if we specialize a bit, let E be a Grothendieck topos that has enough points (that is, such that there is a small set I of inverse image functors E -> Set such that the joint functor E -> S^I is faithful). Then E is equivalent to the category of coalgebras for a certain comonad G defined on S^I = S/I . The comonad G is the called the comonad of GERMS of E. If we specialize even more and take E to be sheaves over a topological space X then I can be taken to be just the set of points of X. Then analysing a coalgebra Ooops, then given a colagebra A for G, it corresponds to a sheaf on X, but out of A we can construct directly the espace etale of the sheaf. The correspondence is good enough to allow one to call the coalgebra A THE espace etale of a sheaf. In particular, a coalgebra is made of a pair (A,h), where A is an object of S/I and h:A -> GA the structural map; A is indeed a set of points defined above I, and the map h is a choice for every point of A of a germ (an equivalence class of sections) which allows to recover the espace etale topology on A. This is what allows us to consider coalgebra structures as espaces etales for any comonad G defined in the right fashion from a Grothendieck topos E. Francois Lamarche +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++