On Tuesday 11 February 2003 16:48, you wrote:
Hello,
I have some questions about the category whose objects are Heyting algebras and whose arrows are Heyting algebra homomorphims.
1) Does this category possess a subobject classifier?
2) Is this category a CCC?
Unless my definition of Heyting algebra is a bit off, I am sure that this (and hence 3) is false. I assume that in a Heyting algebra T does not equal F. This follows the intuitive introduction of Heyting algebras by Moerdijk/MacLane as capturing the algebraic structure of topologies. If that is not the case then disregard the rest of my message. Anyway, under these assumptions, the trivial HA {T,F} is both initial and final. Hence 0 = 1 (= means is iso to). Any CCC with 0 = 1 is trivial. I will leave the diagram chase to you but it can be summarized as follows. Let A be any HA. Then A = A^1 = A^0 = 1. Hope this helps, Bob McGrail
3) Is this category a topos?
It would really be neat if 3) was true because of all kinds of self-reference or infinite regression, e.g. it's Omega would be an internal Heyting algebra, but my guess is "no" to all three.
Regards, Bill Halchin