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