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
.... Dear Bill, The answer is probably in Goldblatt anyway, but the key to it is to realise that morphisms from 1 to Omega are equivalent to subobjects of 1 (i.e., set theoretically, to subsets of a singleton set). Just from your understanding of sets, you should quickly be able to think of two subobjects. It is not difficult to express them in the topos abstraction. There remains the question of whether those subobjects are distinct. There is in fact a topos in which they are the same, but that is pathological behaviour - it is the categorical embodiment of an inconsistent set theory in which true <=> false and 1 = 0. There is (up to equivalence) only one such topos. "The right direction" that you ask for is really to be guided by your set theoretic instincts. You will discover on the way that you need to beware of certain non-constructive aspects of classical mathematics, principally excluded middle and the axiom of choice. Steve Vickers.