Third Scottish Category Theory Seminar

Department of Computer and Information Sciences, University of Strathclyde
Thursday, December 2, 2010

Titles and Abstracts

On Higher Order Algebra
Marcelo Fiore (University of Cambridge)

The purpose of this talk is to give an overview of recent work on a mathematical theory of higher-order algebraic structure. Specifically, I will introduce a conservative extension of universal algebra and equational logic from first to second order that provides a model theory and formal deductive system for languages with variable binding and parameterised metavariables. Mathematical theories encompassed by the framework include the (untyped and typed) lambda calculus, predicate logic, integration, etc. Subsequently, I will consider the subject from the viewpoint of categorical algebra, introducing second-order algebraic theories and functorial models, and establishing correspondences between theories and presentations and between models and algebras. The concept of theory morphism leads to a mathematical definition of syntactic translation that formalises notions such as encodings and transforms in the context of languages with variable binding.

Bohrification: Topos theory and quantum theory
Bas Spitters (University of Nijmegen)

The recently developed technique of Bohrification associates to a (unital) C*-algebra

We will survey this technique, provide a short comparison with the related work by Isham and co-workers, which motivated Bohrification, and use sites and geometric logic to give a concrete external presentation of the internal locale. The points of this locale may be physically interpreted as (partial) measurement outcomes.

Hochas and Minimal Toposes
Peter Johnstone (University of Cambridge)

Recently D. Pataraia introduced the notion of hocha (higher-order cylindric Heyting algebra) in connection with his solution of the problem ”does every Heyting algebra occur as Sub(1) in a topos?”. In this talk we focus on the relationship between hochas and topises: we show that there is an equivalence between (finitary) hochas and toposes satisfying a natural ”minimality” condi- tion, and in particular that the category of finitary hochas is nothing other than the ind-completion of the dual of the free topos.

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