A monadicity theorem for distribution algebras
Marta Bunge – 4 July 2001
If S is an elementary topos and W is its subobject classifier, then W(-):Sop-->S is monadic by Paré's theorem (see also Mikkelsen ). Moreover, there exists an equivalence between Sop and the category of complete atomic Heyting algebras in S (over S, the latter equipped with the forgetful functor and its left adjoint - the free complete atomic Heyting algebra functor).
In  we prove a relative version of Paré's theorem motivated by an interesting interpretation of it in terms of distributions in the sense of Lawvere  and their algebraic duals. The relativization consists in replacing S by a topos E bounded over S, and by replaicing Sop by the category of S-valued distributions on E in the sense of Lawvere . For E=S, with E bounded over S by the identity geometric morphism, the original theorem from  is recovered.
The basic set-up is that of a bounded geometric morphism e:E-->S, whereby E is canonically regarded as an S-indexed category . Further, it is an S-cocomplete S-indexed category, so that the notion of an S-cocontinuous functor m:S-->S is meaningful. Such functors have been identified with S-valued distributions on the S-cocomplete category E.
By a distribution algebra in E (over S)  is meant an S-bicomplete S-atomic Heyting algebra H in E, notions which need to be defined. The terminology employed is suggested from teh contravariant equivalence which exists between (i) distribution algebras H in E over S and (ii) objects of the form m*(W) where m:E-->S is an S-valued distribution on E, m* is its right adjoint, and W is the subobject classifier in S.
There is a "double dualization monad" on E whose category of algebras is equivalent to the opposite of the category of distribution algebras in E over S. Alternatively, the category DS(E) of distribution algebras in E relative to S is monadic over E by means of the forgetful functor U and its left adjoint F.
However, we are seemingly forced to make a hypothesis on E as a topos over S for the monadicity to hold. We are able to prove that the (relative) monadicity theorem holds in two cases: (1) for any topos E which is an essential localization  of a presheaf topos, and (2) when the base topos S is Set.
The monadicity theorem for distribution algebras is in fact only dependent on the existence of a left adjoint F to the forgetful functor U from the category of distribution algebras in E to E, that is on the existence of free distribution algebras. This involves (among other things) the existence of S-cocompletions and needs further investigation.
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