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# Oxford Quantum Talks Archive

Principally-generated modules on a quantale

Isar Stubbe, University of Antwerp
Categories, Logic and Foundations of Physics IV, January 2009, Imperial College London

Let [[$Q$]] be a quantale, and [[$\rm{Mod}(Q)$]] the locally ordered category of [[$Q$]]-modules and [[$Q$]]-module morphisms. Using splittings of idempotents and adjunctions in [[$\rm{Mod}(Q)$]], I shall define the 'principal elements' of a given [[$Q$]]-module; a 'principally generated' [[$Q$]]-module is then one "with enough principal elements". Applying this to a module over the two-element chain, i.e. a complete lattice, will give a familiar notion. I shall explain how, in general, this is related to the theory of ordered (!) sheaves over [[$Q$]]. When [[$Q$]] is moreover involutive, it makes sense to speak of `principally symmetric' [[$Q$]]-modules; this then is related to sheaves over [[$Q$]]. Time permitting, I shall say a word or two on 'Hilbert [[$Q$]]-modules' too.

[video] [streaming video]