On Thu, 24 Oct 2002, Vaughan Pratt wrote:
From: "Al Vilcius" <al.r@vilcius.com>
We have a beautiful notion of subobject classifier true:1-->omega Is there a similar (ie. dual) notion of quotient (co)classifier? - perhaps formulated using pushouts?
This is of course what omega becomes in the opposite of any topos. While the duality principle for ordinary categories makes any separate treatment of the dual case redundant, this is not so clear for practical applications. Has this in fact happened: has anyone gotten any useful mileage out of abstracting CABA to dual toposes as the dual of abstracting Set to toposes, e.g. applications to or generalizations of Boolean-based logics?
My impression is that there are rather few "naturally occurring" categories which have quotient co-classifiers. One reason is that the presence of a subobject classifier forces all subobject lattices in the category to be Heyting algebras (or at least Heyting semilattices -- they needn't actually be lattices); and, whereas this is not such an uncommon phenomenon in concrete categories, lattices of quotients in such categories tend to be co-Heyting algebras rather than Heyting algebras. I'm thinking in particular of such categories as varieties of finitary algebras. In my paper "Collapsed toposes and cartesian closed varieties" (J. Algebra 129, 1990), I characterized the varieties which have subobject clasifiers, and they are exactly those in which subobject lattices are distributive. There are well-known algebraic conditions for congruence lattices (dual to lattices of regular quatients) to be distributive. However, in answer to a question from Fred Linton, I once found a proof that no non-degenerate finitary variety (indeed, no variety with rank) possesses a quotient co-classifier. That proof was never written up, so perhaps this would be an occasion to publicize it. First of all, recall that in a variety the class of all epimorphisms needn't coincide with that of regular epimorphisms; but if there is a co-classifier for epimorphisms then all epimorphisms must be regular, and so we might as well work with regular epimorphisms (or equivalently with congruences). Now suppose we have a congruence co-classifier \mho in a variety with rank. (\mho is an upside-down \Omega, by the way.) Since the variety is locally presentable, \mho will be \kappa-presentable for some \kappa (which may as well be greater than or equal to the rank of the variety). Thus if F is a free algebra on more than \kappa generators, any homomorphism \mho \to F must factor through the free subalgebra on some subset of the generators of cardinality less than \kappa. It is thus easy to see that the quotient map from F to the free algebra on one generator, obtained by setting all the generators of F equal to one another, cannot be obtained as a pushout of a quotient of \mho along any such homomorphism.
What does "cotopos" mean for V.L. Vasyukov, as in his "Developing Tarski: a cotopos of theories" (Logical Investigations. Vol 3, Moscow, 1995.)?
I have a vague memory of having once looked at this paper (though I can't now find a copy of it, and -- rather to my surprise -- it didn't get into the bibliography of "Sketches of an Elephant"). If it's the paper I'm thinking of, then it was so sloppily written that I never managed to find out exactly what the author meant by "cotopos". Peter Johnstone 25-Oct-2002 16:27:20 -0300,1225;000000000001-00000000