Math and Stats - ATCAT seminar - "The structure of ω-complete effect monoids"

Speaker: Abraham Westerbaan (Dalhousie University)

Abstract:
Effect monoids are a generalisation of {0,1} and [0,1], of Boolean algebras and the unit intervals [0,1] of commutative unital C*-algebras.  Effect monoids appeared naturally in the study of effectuses: a type of category with finite coproducts 0, + and a final object 1 designed to reason about states s: 1⟶X and predicates p: X⟶1+1.  When composing such a state and predicate, one gets a morphisms 1⟶1+1 that should be thought of as the probability that the predicate p holds in state s.  It’s these morphisms 1⟶1+1 called scalars that form an effect monoid.

Vanilla effect monoids are lousy structures:  not much can be defined with(in) them, or be proven about them, while counter examples hard to find.  This changes dramatically when the axiom of ω-completeness is added (that every ascending sequence in the effect monoid has a supremum.)  Suddenly a rich and well-behaved structure emerges including division, lattice operations, and an abundance of idempotents.  So well-behaved, in fact, that every ω-complete effect monoid can be represented as subspace of the continuous functions C(X,[0,1]) on a basically disconnected compact Hausdorff space X.  For directed complete effect monoids we even get a proper categorical duality.

This is based on joint work with Bas Westerbaan and John van de Wetering:  https://arxiv.org/abs/1912.10040

Category

Lectures, Seminars

Time

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Location

https://us02web.zoom.us/j/81872756284?pwd=VGVQcW8vRmFXK1hvTk5UZURGSXg5UT09 Meeting ID: 818 7275 6284 Passcode: ATCAT2020

Contact

Ellen Lynch