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From: Robert McGrail <mcgrail@bard.edu>
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Subject: Re: Category of Heyting Algebras
Date: Wed, 12 Feb 2003 12:20:09 -0500
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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