Quantales are complete lattices equipped with an associative binary operation \(\circ\) which preserves suprema in both variables. Quantales arise in many areas of mathematics and recently have provoked much interest in theoretical computer science and linear logic. (For an overview of quantale theory, see the reviewer’s book: Quantales and their applications (1990; Zbl 0703.06007).)
The original definition of a quantale was much more restrictive, requiring \(\circ\) to be idempotent, thus allowing the view of quantales as “non-commutative locales” (or, more precisely frames, if one wants the morphisms to preserve \(\circ\) and sups). In this article, the authors use this definition and also they relax the requirement of sup- preservation to one of the two variables. Thus, for a right quantale \({\mathcal Q}\), they require the operation \(\circ\) to be idempotent and sup- preserving on the right. \({\mathcal Q}\) is also right-sided, which means that \(a\circ\tau=a\) for all \(a\in{\mathcal Q}\), where \(\tau\) is the top element of \({\mathcal Q}\). The main thrust of the article is to consider the notion of sheaf over such a quantale with an eye towards generalizing the construction of the topos \(Sh(L)\) of sheaves on a locale \(L\). It is well known that there are equivalent ways of arriving at the notion of a sheaf. There is the usual functorial description as a presheaf satisfying the sheaf “pasting” condition. There is also the description in terms of locales étale over \(L\), as well as the notion of sets equipped with an \(L\)-valued equality. The authors prove that all of these constructions generalize to right quantales with the resultant categories all equivalent. They then proceed to investigate some of the categorical properties of sheaves on a right quantale, which, while not being a topos, does have some nice properties.
One still awaits a satisfactory notion of sheaf for the more general notion of quantale, with the idempotence of \(\circ\) dropped. One possibly fruitful approach might be provided by the work of Walters, who described sheaves on a locale \(L\) as certain Cauchy complete categories enriched in a suitable locally partially ordered bicategory constructed from \(L\) [R. F. C. Walters, J. Pure Appl. Algebra 24, 95-102 (1982; Zbl 0497.18016)].