From: Robert McGrail <mcgrail@bard.edu>
To: categories@mta.ca
Subject: Re: Category of Heyting Algebras
Date: Wed, 12 Feb 2003 12:20:09 -0500 [thread overview]
Message-ID: <200302121220.09792.mcgrail@bard.edu> (raw)
In-Reply-To: <20030211214817.55877.qmail@web12203.mail.yahoo.com>
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
next prev parent reply other threads:[~2003-02-12 17:20 UTC|newest]
Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top
2003-02-11 21:48 Galchin Vasili
2003-02-12 17:03 ` Prof. Peter Johnstone
2003-02-14 3:32 ` Robin Cockett
2003-02-12 17:20 ` Robert McGrail [this message]
2003-02-12 19:31 ` Toby Bartels
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