University of Sydney Algebra Seminar
Richard Garner
Friday 26 August, 12-1pm, Place: Carslaw 173
Cartesian closed varieties
A (universal algebraist's) variety is the category of models of an equational theory: for example, we have the variety of groups, of rings, of meet-semilattices, of Boolean algebras, and so on. As a category, any variety has cartesian products. It is natural to ask when these products have an associated internal hom, i.e., when a variety is a cartesian closed category. Peter Johnstone gave a syntactic answer to this question in the 1990s. In this talk, we describe a corresponding semantic answer; any such variety is made up of sets endowed with actions by a monoid M and a Boolean algebra B which interact in a specific way. It turns out that the monoid--Boolean algebra pairs arising in this context are related to structures in non-commutative geometry. We will illustrate this with some examples coming from the theory of combinatorial C*-algebras.