Galchin Vasili wrote:
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".
The element true of Hom(1,Omega) corresponds to the subobject id: 1 -> 1 of 1. To get another subobject of 1, use the unique arrow 0 -> 1; the corresponding element of Hom(1,Omega) is false. However, it's still possible that true = false, in which case true is the only element of Hom(1,Omega). This is a particularly degenerate case, but the definition of a topos doesn't rule it out. The trivial category (one object and one morphism) is an example of such a degenerate topos. You can even prove the converse. If true = false, then 0 = 1, so x = x^1 = x^0 = 1. Here "=" means a unique isomorphism, so the topos is equivalent to the trivial category. -- Toby