I have a conjecture: the dual of exponential is boring in any category. Is there a counterexample to this conjecture? If not how can we prove it?
Well, if boring means non-trivial there are examples, namely opposites of cartesian closed categories. E.g. Set^op equivalent to CABA (complete atomic boolean algebras). E.g. for a topological space X the poset C(X) of closed subsets of X ordered by set inclusion is an example. There conegation ~A is the closure of the complement of A and A \cap ~A is the border of A. This received some attention as models of "dialectical logic". There are also biHeyting algebras meaning that A and A^op are Heyting algebras. Subobject lattices of objects in presheaf toposes are examples of this as observed by Lawvere. But if you mean by coexponential A x (_) having a left adjoint then in presence of a terminal object 1 this means A is isomorphic to 1 and indeed they are trivial. Thomas [For admin and other information see: http://www.mta.ca/~cat-dist/ ]