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? One imagines that, just as natural numbers inhabit the natural number object of a topos T, propositions would inhabit the proposition objects of a dual topos T\op, defined simply as the power objects Omega^X in T, where arbitrary object X of T takes on the role of propositional variables object. ("Proposition object" here is intended as the topos-theoretic counterpart of free Boolean algebra, or rather free CABA.) Or is there a better notion of proposition object for a dual topos? What does "cotopos" mean for V.L. Vasyukov, as in his "Developing Tarski: a cotopos of theories" (Logical Investigations. Vol 3, Moscow, 1995.)? Vaughan Pratt 25-Oct-2002 14:53:17 -0300,7162;000000000000-00000000