Re: Category of Heyting Algebras

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

On Tue, 11 Feb 2003, Galchin Vasili 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?
>
>  3) Is this category a topos?
>
The category of Heyting algebras has no hope of being cartesian closed
because its initial object (the free HA on one generator) is not
strict initial. It doesn't have a subobject classifier either, because
the theory of Heyting algebras doesn't have enough unary operations
to satisfy the conditions of Theorem 1.3 in my paper "Collapsed
Toposes and Cartesian Closed Varieties" (J. Algebra 129, 1990).

On the other hand, the terminal object in the category of Heyting
algebras is strict, which suggests that the dual of the category
might come rather closer to being a topos (although, by an observation
which I posted a couple of months ago, it can't have a subobject
classifier). Indeed, the dual of (finitely-presented Heyting
algebras) is remarkably well-behaved, as shown by Silvio Ghilardi
and Marek Zawadowski ("A Sheaf Representation and Duality for
Finitely Presented Heyting Algebras", J.Symbolic Logic 60, 1995):
they identified a particular topos in which it embeds (non-fully,
but conservatively) as a subcategory closed under finite limits,
images and universal quantification.

Peter Johnstone