# Modules over quantales and quantaloids

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

## Idea

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. The special case (when the quantale is unital, commutative and hence a locale) is the case of modules over a locale (localic modules), and the multiobject version are the quantaloid modules.

## 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

Created on April 9, 2014 at 03:33:40. See the history of this page for a list of all contributions to it.