Étale maps of quantales. (English) Zbl 0687.06005

Berni-Canani, U.; Borceux, F.; Succi-Cruciani, R.; Van den Bossche, G.

Bull. Soc. Math. Belg., Sér. A 41, No. 2, 195-218 (1989).

Quantales form an equational class of complete multiplicative lattices. A motivating example is the lattice of closed right ideals of a \(C^*\)- algebra. Commutative quantales are locales, i.e. generalized open set lattices of topological spaces. Quantales are meant as one possible approach to non-commutative topology. The present paper contributes to this program by introducing a tensor product and étale maps of quantales. Étale maps are then used for defining sheaves on quantales. The first definition of sheaves on quantales was given by M. Nawaz via quantic sets in 1985. F. Borceux and G. Van den Bossche used sites [Order 3, 61-87 (1986; Zbl 0595.18003)].

Reviewer: J.Rosický

MSC:

06B23	Complete lattices, completions
06F05	Ordered semigroups and monoids
18B30	Categories of topological spaces and continuous mappings (MSC2010)

Keywords:

equational class of complete multiplicative lattices; lattice of closed right ideals of a \(C^*\)-algebra; quantales; locales; generalized open set lattices of topological spaces; non-commutative topology; tensor product; étale maps; sheaves on quantales

Citations:

Zbl 0595.18003