It seems that you can modify the construction of the localic Diaconescu cover to get an open surjective _filtered_ cover of a Grothendieck topos Sh(C). Instead of using the category String(C) of strings in C, you could construct the category Tree(C) of finite, rooted, binary trees in C. If given c and d in C you can find e such that e --> c and e --> d then Tree(C) is a poset with binary upper bounds, i.e. a filtered category. Could anyone provide a reference for such a construction? Grateful for any help, Jonas Eliasson ------------------------------------------ | Jonas Eliasson | | Department of Mathematics | | Uppsala University | | Sweden | | E-mail: jonase@math.uu.se | | Homepage: http://www.math.uu.se/~jonase/ | ------------------------------------------