R. J. Wood

Constructive complete distributivity. I

The relationships, in many cases equivalences, between lattice distributivity, adjunction and continuity have been studied by many authors, for example [ 1, 3–8, 12, 13, 15, 17–20, 22, 23 ]. Very roughly, we refer to the following circle of ideas. Let L be an ordered set, and L a class of subsets of L , and suppose that L has a supremum for each element in L . We might say that L has -sups. The ‘distributivity’ we refer to is that of infs over -sups. The ‘adjunction’ is that given by a left adjoint to the map V: L→L . Now the latter has a left adjoint if and only if it preserves infs, and this means roughly that the -sup of an intersection is an inf of -sups. When one does succeed in identifying the -sup of an intersection as a -sup of infs, one has an instance of distributivity.

Indexed categories and their applications

Abstract families and the adjoint functor theorems.- V-indexed categories.- Algebraic theories in toposes.- Coequalizers in algebras for an internal type.

Constructive complete distributivity IV

A complete latticeL isconstructively completely distributive, (CCD), when the sup arrow from down-closed subobjects ofL toL has a left adjoint. The Karoubian envelope of the bicategory of relations is biequivalent to the bicategory of (CCD) lattices and sup-preserving arrows. There is a restriction to order ideals and “totally algebraic” lattices. Both biequivalences have left exact versions. As applications we characterize projective sup lattices and recover a known characterization of projective frames. Also, the known characterization of nuclear sup lattices in set as completely distributive lattices is extended to yet another characterization of (CCD) lattices in a topos.

Distributive laws and factorization

Abstract This article shows that the distributive laws of Beck in the bicategory of sets and matrices, wherein monads are categories, determine strict factorization systems on their composite monads. Conversely, it is shown that strict factorization systems on categories give rise to distributive laws. Moreover, these processes are shown to be mutually inverse in a precise sense. Strict factorization systems are shown to be the strict algebras for the 2-monad (−) 2 on the 2-category of categories. Further, an extension of the distributive law concept provides a correspondence with the classical factorization systems.

A basic distributive law

Abstract We pursue distributive laws between monads, particularly in the context of KZ-doctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect to the subobject classifier) on ordered sets.

An Adjoint Characterization of the Category of Sets

If a category B with Yoneda embedding Y: B CAT(BOP, set) has an adjoint string, U H V H W H X H Y, then B is equivalent to set.

Proarrows and Cofibrations

We showed earlier that for the proarrow equipment ( )∗: TOP→TOPLEXco, the codomain is equivalent to codiscrete cofibrations in the domain (i.e. TOPLEXco∼CODCOFIB TOP). Here we show that M∼CODCOFIB K for any proarrow equipment ( )∗:K→M satisfying a finitary exactness axiom. We give applications to topoi relative to a base and Grothendieck topoi.

Boundedness and Complete Distributivity

We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZ-doctrine for bounded suprema is of some independent interest and a few results about it are given. The 2-category of ordered sets admitting bounded suprema over which inhabited (classically non-empty) infima distribute is shown to be bi-equivalent to a 2-category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the non-negative reals.

Groupoidal completely distributive lattices

Abstract The Adamek and Pedicchio proof that top op is a quasi-variety is adapted to show that the opposite of the category of pre-ordered sets is also a quasi-variety. The constructive proof given requires a description of power objects in terms of (constructively) completely distributive lattices and such a description is provided by the Carboni and Walters notion of “groupoidal” object in a cartesian bicategory.

An extension of the regular completion

Carbonis regular completion doctrine is extended to a KZ-doctrine on a 2-category whose objects are all categories and whose arrows are functors which preserve kernel arrows. The algebras for the extended doctrine are categories with regular factorizations in which regular epimorphisms are closed under composition