[note from moderator: before I had a chance to post this response, several other readers also pointed out the degeneracy involved, so those will not be posted. Regards, Bob] On Mon, 2 Dec 2002, Galchin Vasili wrote:
Fact: In a topos, Hom (1, omega) is a Heyting algebra.
Fact: A Heyting algebra is a CCC (closed catesian category).
Question: is there a topos T where Hom (1, omega) itself is a topos or equivalently
has a subobject classifier (in addition to being a CCC)?
Question: If there is no such topos T, where can I find a proof that no such topos
exists?
I'm not sure what Bill means by this question -- it seems to contain a category mistake (as the philosophers would say). Hom (1,\Omega) is always a partial order, but a topos can never be a partial order unless it is degenerate. Perhaps he is groping towards the notion of quasitopos, which is a common generalization of the notions of topos and of Heyting algebra -- see section A2.6 of "Sketches of an Elephant", or alternatively Oswald Wyler's book "Lecture Notes on Topoi and Quasitopoi". Peter Johnstone