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Mike Stay (Aug 27 2020 at 18:14):I've heard it claimed that there's a representation theorem that says that any quantale is equivalent to the powerset of the underlying set of a monoid. If we categorify to and a monoid to a monoidal category, we get a topos of presheaves on a monoidal category. Can someone point me at literature that explores this relationship?
It also seems like we can generalize powerset to in an arbitrary topos; can we generalize presheaves on a monoidal category to V-presheaves on a V-enriched monoidal category?
John Baez (Aug 27 2020 at 22:20):I've heard it claimed that there's a representation theorem that says that any quantale is equivalent to the powerset of the underlying set of a monoid.
Who's running around claiming this? I have trouble believing it.
Nathanael Arkor (Aug 27 2020 at 22:24):I imagine the result @Mike Stay is referring to is Brown–Gurr's A representation theorem for quantales.
We define a relational quantale to be a quantale whose elements are relations on a set A, ordered by inclusion and forming a monoid under relational composition. Such quantales have been studied in several areas of theoretical computer science, and constitute a sound and complete class of models for non-commutative linear logic. We show that every quantale is isomorphic to a relational quantale, and investigate the classification of quantales according to properties of their representations as relational quantales.
Mike Stay (Aug 27 2020 at 22:33):@Nathanael Arkor Thanks!
Nathanael Arkor (Aug 27 2020 at 22:34):Nishizawa–Furusawa's Relational Representation Theorem for Powerset Quantales has another representation result that has some nicer properties.
Nathanael Arkor (Aug 27 2020 at 22:36):Note that these are restricted subclasses of quantales (e.g. the completely coprime algebraic quantales), which may not be so evident to express as Bénabou cosmoi.