Quantales and their sheaves. (English) Zbl 0595.18003

Borceux, Francis; van den Bossche, Gilberte

Order 3, 61-87 (1986).

A quantale, as defined by the authors, is the non-commutative analogue of a locale. Thus, a quantale Q is a complete lattice with a binary operation \(Q\times Q\to^{\&}Q\), which is associative, idempotent, and sup-preserving in each variable. Elements of Q are taken to be right- sided, i.e., a & 1\(=a\) for all \(a\in Q\), where 1 denotes the top element of Q. An element is called two-sided if furthermore 1 & a\(=a\). If b is two-sided, then a & b\(=a\wedge b\) for all \(a\in Q\), and it follows that the two-sided elements of Q form a locale. It should be pointed out that a different notion of quantale has been considered in other contexts, where the requirement of idempotence is dropped.
The goal of this paper is to associate to each quantale a Grothendieck topos of sheaves (not necessarily localic). It is hoped that this will provide a framework for investigating the representation theory of non- commutative rings and algebras. The paper begins by developing some of the basic theory of quantales and their morphisms, as well as generalizing certain concepts (such as points) and certain constructions (such as coproducts) from the localic case. In developing the notion of sheaf on a quantale Q, the non-commutativity comes into play in that given \(a\leq b\) in Q, there are two restriction maps \(\downarrow a\to \downarrow b\), right and left restriction, which need not coincide. In fact, right restriction is well defined if and only if a & b\(=b.\)
The category \({\mathcal Q}\) is formed, whose objects are \(\downarrow a\) for \(a\in Q\) and whose morphisms are the left restrictions and the right restrictions, when they happen to be defined. \({\mathcal Q}\) is made into a site by defining a Grothendieck topology, which essentially requires coverings to have ”sufficiently many” right restrictions. This leads to the desired topos of sheaves. If Q is a locale, the usual construction of sheaves on a locale is recovered. sh(\({\mathcal Q})\) has the additional feature that the locale of subobjects of an object comes equipped with an additional quantale structure extending that of the original quantale. There is a surjective strict morphism Sub(1)\(\to Q\) of quantales, where Sub(1) is endowed with this additional structure. Thus, in addition to the internal logic of the topos, there is the ”quantum” logic of \({\mathcal Q}\) reflected in sh(\({\mathcal Q}).\)
As a final thought, it would be nice if these ideas could be even further generalized by dropping the requirement of idempotence. There are many interesting quantales without an idempotent operation, such as the quantale of all right ideals of a non-commutative ring. It might prove useful to associate to such a quantale a category of ”sheaves” (it might no longer be a topos, but it should at least be a closed category).

Reviewer: K.I.Rosenthal

MSC:

18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
06D99	Distributive lattices
81P10	Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
18F10	Grothendieck topologies and Grothendieck topoi
18B25	Topoi
03G12	Quantum logic

Keywords:

quantum logic; quantale; locale; complete lattice; binary operation; two- sided elements; Grothendieck topos of sheaves; Grothendieck topology

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