Category of Heyting Algebras

Category of Heyting Algebras

[Galchin Vasili]

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?

 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

Re: Category of Heyting Algebras

[Prof. Peter Johnstone]

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

Re: Category of Heyting Algebras

[Robert McGrail]

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

Re: Category of Heyting Algebras

[Toby Bartels]

Robert McGrail wrote:

>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.

Probably there are people that put this in the definition --
the same people that require 0 != 1 in any ring,
or that a topological space not be empty,
or (for that matter) that a CCC not be trivial.
But if you're not one of those people,
then the Heyting algebra {*} captures the algebraic structure of
the (unique topology on the) empty space.

>Anyway, under these assumptions, the trivial HA {T,F} is both initial and
>final.

So with my definition, {T,F} is initial but {*} is final.
But even with yours, I don't believe that {T,F} becomes final.
There are 2 homomorphisms to it from the power set of {T,F}
(as a Boolean algebra, which is a special kind of Heyting algebra).
I don't think that your category has a final object
(any more than the category of nontrivial rings does,
nor the category of nonempty spaces has an initial object).


-- Toby

Re: Category of Heyting Algebras

[Robin Cockett]

Further to Peter's remark that the opposite of the category finitely
presented Heyting algebras is rather nice ... one particular sense in
which it is nice is that it is a lextensive category!  So -- to
seemingly contradict Peter :-) -- it does have a partial map classifier
(but of course not for all monics)!

The fact that it is a lextensive category can be obtained, by checking
some Heyting algebra identities, from

"Conditional Control is not quite Categorical Control"
IV Higher Order Workshop, Banff 1990, Workshops in Computing
(Springer-Verlag) 190-217 (1991)

where the general question of when the opposite of an algebraic theory
is extensive is answered.

Any extensive category can be fully and faithfully embedded in a topos
so as to preserve sums and limits ... so the ability to embed the
opposite of Heyting algebra "nicely" into a topos can also be
read from these results.

-robin

P.S.Whether this embedding has the other logical properties
mentioned is, of course, another question ...


On 12 Feb, Prof. Peter Johnstone wrote:
> On Tue, 11 Feb 2003, Galchin Vasili wrote:
>
>>      I have some questions about the category whose objects are Heyting
>> algebras and whose arrows are Heyting algebra homomorphims.
>>
>
> 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
>
>
>
>