Constructive complete distributivity. I

Abstract

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.

Totally distributive toposes

“…The aim of this paper is to establish certain connections between the work of Marmolejo, Rosebrugh, and Wood [14,13] on totally distributive categories and two other bodies of work on distinct topics: Firstly, that of Johnstone and Joyal [4,7] on injective toposes and continuous categories, and secondly, that of Kelly-Lawvere [8] and Kennett-Riehl-Roy-Zaks [9] on essential localizations and essential subtoposes. One of our observations, 1.5.9 (2), when taken together with a theorem of Kelly-Lawvere which we recall in 1.5.6, yields a concrete combinatorial description of all totally distributive categories with a small set of generators.…”

“…A poset E is a constructively completely distributive lattice [2], or ccd lattice, if there exist adjunctions…”

“…The existence of the left adjoint ∨ of ↓ is equivalent to the cocompleteness of E, i.e. the condition that E be a complete lattice, and if such a map ∨ exists, it necessarily sends each down-closed subset to its join in E. In the presence of the axiom of choice, a poset is a ccd lattice iff it is a completely distributive lattice in the usual sense [2].…”

The General Theory of Diads

“…We give the definitions in Section 3.1, then in Section 3.2, we find the category of dialgebras for the examples of diads from Section 2.4. In this way, we construct the category of constructively completely distributive lattices, studied in [2,9,10], and [11], etc. as a category of dialgebras for a diad.…”

“…The concept of constructive complete distributivity is introduced in [2], and studied further in [9][10][11], ... In a boolean topos, it is equivalent to complete distributivity.…”