On Tue, 20 Nov 2007, Lutz Schroeder wrote:
Is the Heyting algebra of global elements of the classifier in an elementary topos always complete?
The answer is no: any Boolean algebra, complete or not, can occur as Sub(1) in a topos (see Exercise 9.11 in my old Topos Theory book), and any quotient of a complete Heyting algebra (by a finitary Heyting congruence -- such quotients needn't be complete) can occur, by use of the filterpower construction (cf. my paper with Murray Adelman "Serre classes for toposes", Bull.Austral.Math.Soc. 25 (1982), 103-115). There are examples due to Peter Freyd of Heyting algebras which can't occur as Sub(1) in a topos generated by subobjects of 1. For a long time it was an open problem whether any Heyting algebra can occur as Sub(1) in a topos (without the restriction on generators): Dito Pataraia has recently announced a positive solution to this problem. I have heard a seminar talk about his solution, and seen half of a preprint, but haven't yet managed to understand the other half of his construction. Peter Johnstone