Abstract

The talk consist of two separate but connected parts. The first part is joint work with Dion Coumans (Nijmegen). It describes interrelatedness between:

1. algebraic structure on sets of scalars,
2. properties of monads associated with such sets of scalars, and
3. structure in categories (esp. Lawvere theories) associated with these monads.

These interrelations will be expressed in terms of ``triangles of adjunctions’’, involving for instance various kinds of monoids (non-commutative, commutative, involutive) and semirings as scalars. It will be shown to which kind of monads and categories these algebraic structures correspond via adjunctions.

The second part will investigate extensions of these results to probabilities as scalars. It involves convex sets, effect algebras, and a new class of functors that we call `convex functors’; they include what are usually called probablity distribution functors. The relationships take the form of three adjunctions. Two of these three are `dual’ adjunctions for convex sets, one time with the Boolean truth values {0,1} as dualising object, and one time with the probablity values [0,1] of the unit interval. The third one is a new adjunction between effect algebras and convex functors, forming the first step towards a ``triangle of adjunctions’’ in this probabilistic area.