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 [15]). 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 [8] 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 [13] 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 [13]. For E=S, with E bounded over S by the identity geometric morphism, the original theorem from [17] 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 [18]. 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) [8] 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 [14] 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. References M. Barr and R. Paré, Molecular toposes, J. Pure Appl. Alg. 17(1980) 127-152. J. Bénabou, Fibred categories and the foundations of naive category theory, J. Symbolic Logic 50(1985) 10-37. G.J. Bird, Limits in 2-categories of locally presented categories, Ph.D. thesis, University of Sydney, 1984. M. Bunge, Cosheaves and distributions on toposes, Alg. Univ. 34(1995) 233-249. M. Bunge and A. Carboni, The symmetric topos, J. Pure Appl. Alg. 105(1995) 233-249. M. Bunge and J. Funk, Spreads and the symmetric topos, J. Pure Appl. Alg. 113(1996) 1-38. M. Bunge and J. Funk, Spreads and the symmetric topos II, J. Pure Appl. Alg. 130(1998) 49-84. M. Bunge, J. Funk, M. Jibladze, T. Streicher, Distribution algebras and duality, Adv. Math. 156(2000) 133-155. J. Funk, Descent for cocomplete categories, Ph.D. thesis, McGill University, 1990. J. Funk, The display locale of a cosheaf, Cahiers de Top. et Géo. Diff. Cat. 36(1995) 53-93. P.T. Johnstone, Open maps of toposes, Manuscripta Math. 31(1980) 217-247. F.W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proc. Symposia in Pure Mathematics of the American Math. Soc. 17 1-14. F.W. Lawvere, Categories of space and quantity, in: J. Echeverria et al. eds. The Space of Mathematics, W. de Gruyter, Berlin (1992) 14-30. F.W. Lawvere and G.M. Kelly, On the complete lattice of essential localizations, Bull. Soc. Math. Belg. XLI(1989) 289-319. C.J. Mikkelsen, Lattice theoretic and logical aspects of elementary toposes, Ph.D. thesis, Âarhus Universiteit, Matematisk Institut, Various Publications Series 25, 1976. J.-L. Moens, Charactérisation des topos de faisceaux sur un site interne à un topos, Ph.D. thesis, Univ. Cath. Louvain-la-Neuve (1982). R. Paré, Colimits in topoi, Bull. Amer. Math. Soc. 80(1974) 556-561. R. Paré and D. Schumacher, Abstract families and the adjoint functor theorems, in: P.T. Johnstone and R. Paré, eds. Indexed categories and their applications Springer LNM 661(1978) 1-125. A.M. Pitts, On product and change of base for toposes, Cahiers de Top. et Géo Diff. Cat. 26(1975) 43-61.