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". Can somebody point me in the right direction? Thank you, Bill Halchin
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
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.
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
participants (4)
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Dr. P.T. Johnstone -
Galchin Vasili -
S.J.Vickers＠open.ac.uk -
Toby Bartels