Given a sheaf on a topological space, one can construct the espace etal\'e of that sheaf and recover the sheaf as the sheaf of sections. It would be nice if there were a similar construction for sites. Does anyone know how to do it? Allan Adler ara@altdorf.ai.mit.edu +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Doesn't the following construction work for locales, even without points? Take all sections over arbitrary opens as generators. If s is a section over an open U and if U is a join of opens U(i), then demand s to be the join of the restrictions of s to the U(i). This should give a suitable notion of "locale etal" in terms of generators and relations. If not, why not? Greetings Reinhard +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Reinhard Boerger asks a question of the following sort: This works for spaces why not for locales? To find an answer to all such questions first try whatever it is on a complete atomless boolean algebra. +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
There is a simple notion of what is a local homeomorphism of locales, along with the theorem: given a locale L then the full subcategory of Loc/L whose objects are local homeos is equivalent to Sheaves on L. All this can be found in Joyal-Tierney monograph "An extension of the Galois theory of Grothendieck". The definition of local homeo extends to toposes. But then the theorem becomes trivial: if E->F is a local homeo of toposes then there is an object I of F such that E is equivalent to F/I. If we want results for toposes that are more concrete and less trivial Peter Freyd's suggestion seems the way to go: replace sets by complete boolean algebras^op and use Barr's theorem that says that to any Grothendieck topos is covered (in the sense of having a surjective geometric morphism to it) by a topos of sheaves over a CBA. F Lamarche +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (4)
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Allan Adler -
Francois Lamarche -
pjf＠saul.cis.upenn.edu -
Reinhard Boerger