On Thu, 5 Sep 2002, Galchin Vasili wrote:
Hello CT Community,
I am continuing my study of topoi by reading Goldblatt. In an arbitrary topos E, can we demonstrate that Hom (1, omega) contains any arrows/morphisms other than "true"? I suspect the answer is yes because Hom (1, omega) is a Heyting algebra. Hence, it must have a least element ("true" is the greatest element). I just don't know how to to prove that Hom (1, omega) contains more elements than "true".
That's because it can't be proved: the topos axioms allow the possibility of degeneracy. The category with one object and one morphism is a topos; in it, \Omega = 1 and \top is the only morphism 1 --> \Omega. If you add the requirement that the topos should be non-degenerate (i.e., not equivalent to this example), then \top is not equal to \bot; indeed, this inequality is the easiest way of expressing the condition that a topos is non-degenerate. Peter Johnstone