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Quantal sets and sheaves over quantales. (English) Zbl 0837.18003

Lith. Math. J. 34, No. 1, 8-29 (1994) and Liet. Mat. Rink. 34, No. 1, 9-31 (1994).
Quantales were rediscovered in the 1980’s by C. Mulvey (having been considered in the 1930’s by Ward and Dilworth under the name “residuated lattice”) as a generalization of frames, which are the commutative idempotent 2-sided quantales. [See the reviewer’s book “Quantales and their applications”, Pitman Res. Notes Math. Ser. 234, Longman Scientific & Technical, New York (1990; Zbl 0703.06007) for a detailed look at quantales.] Given the fundamental role played by the theory of sheaves on a frame in categorical logic, ring representation theory etc., it is natural to try to develop the notion of a sheaf on a quantale. There have been various attempts to do so by Borceux, van den Bossche, Cruciani and others focusing on the notion of a quantal set, that is a set valued in a quantale, and exploiting the equivalence of Heyting-valued sets and sheaves in the case of frames. These attempts assumed the idempotence on the quantale \(Q\) in question, rendering the theory somewhat unsatisfactory in the sense that there are many interesting non-idempotent quantales.
In the article under review, the author develops a theory of quantal sets, which while placing some restrictions on \(Q\), allows for a more general class of \(Q\) to be considered. This theory can be viewed as a non-commutative extension of the work of U. Höhle [\(M\)-valued sets and sheaves over integral commutative \(CL\)-monoids, in: Appl. of category theory to fuzzy subsets, Theory Decis. Libr., Ser. B 14, 34-72 (1992; Zbl 0766.03037)].
After introducing the appropriate definition of \(Q\)-set, the concept of “singleton” is developed, which allows one to define a monad on the category of \(Q\)-sets, assigning to every (separated) \(Q\)-set its space of singletons. The Eilenberg-Moore algebras for this monad then give rise to the notion of sheaf on \(Q\). The paper concludes with some discussion comparing this notion of sheaf with some of the earlier notions alluded to above.

MSC:

18B35 Preorders, orders, domains and lattices (viewed as categories)
06F05 Ordered semigroups and monoids
03G30 Categorical logic, topoi
03G25 Other algebras related to logic
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18C20 Eilenberg-Moore and Kleisli constructions for monads
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