Constructive Complete Distributivity II

A complete lattice L is constructively completely distributive CCD L if the sup map de ned on down closed subobjects has a left adjoint It was known that in boolean toposes CCD L is equivalent to CCD L We show here that the latter property for all L su ciently for characterizes boolean toposes Research partially supported by grants from NSERC Canada Diagrams typeset using catmac Introduction The notion of constructive complete distributivity for a complete ordered set L CCD L was de ned in to mean the existence of a left adjoint to W DL L where DL is the set of down closed subsets of L ordered by inclusion Terminology springs from i the fact that CCD L is equivalent to S DL n S j S S o n fT S j S Sg j T S o which becomes the condition for complete distributivity of L CD L if D is replaced by P the power set functor and ii the following Theorem ac CD CCD where ac denotes the axiom of choice Somewhat less cryptically the CCD condition is the restriction of the more familiar CD to down closed subsets it implies the stronger condition in the presence of the axiom of choice which in turn is implied by identifying the two notions of distributivity Thus in classsical lattice theory with the axiom of choice freely assumed it would appear that CCD is simply an easier condition to verify than CD However a considerable body of literature has explained the value of doing Mathematics in a topos and in this more general context the very de nition above and Theorem suggest that CCD is the relevant notion Indeed in any topos PX is CCD for all X while the statement PX is CD for all X is equivalent to choice The power objects PX in a topos are not in general boolean algebras and it transpires that booleaness of the topos of sets does make a sub stantial di erence to the theory of constructive complete distributivity For a topos E we write boo E to indicate that PX is boolean for all X in E We write CCD L for CCD L With this notation Theorem of becomes boo CCD CCD Somewhat contrary to intuition the hypothesis is necessary In this note we prove