nLab module over a quantale

Modules over quantales and quantaloids

(People use also expressions: quantale module, quantic module)

Idea

A quantale is an analogue of a noncommutative ring and a noncommutative generalization of a locale; it is a semigroup in the monoidal category of sup-lattices; therefore, for a quantale $Q$, a left $Q$-module should be a sup-lattice $M$ together with an action $Q\otimes M\to M$ satisfying the usual axioms. In the special case when the quantale is unital and commutative and hence a locale, such a module is called a “localic module”, or module over a locale. The multiobject generalization is called a quantaloid module.

References

• Isar Stubbe, Q-modules are Q-suplattices, Theory and Appl. of Cat. 19, 2007, No. 4, pp 50-60 abs
• Pedro Resende, Groupoid sheaves as quantale sheaves, J. Pure Appl. Algebra 216 (2012), 41–70; arxiv/0807.4848

  doi

• Hans Heymans, Sheaves on quantales as generalized metric spaces, Doctor thesis, Universiteit Antwerpen 2010 pdf

The localic case is studied in

• André Joyal, M. Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.
• Pedro Resende, E. Rodrigues, Sheaves as modules, Appl. Categ. Structures 18 (2010) 199-217; arXiv:0711.4401