Eighth Scottish Category Theory Seminar

International Centre for Mathematical Sciences, Edinburgh
Friday 29th November 2013

Titles and Abstracts

Higher toposes of laws of motion
Urs Schreiber (Radboud University Nijmegen)

In the 90s William Lawvere tried to extract the abstract essence of differential geometry by adding a handful of axioms to topos theory.. An often forgotten fact about the resulting "synthetic differential geometry" is that Lawvere had been all motivated by the more far-reaching desire of axiomatizing what physicists call "classical field theory", whence the title of his seminal "Toposes of laws of motion" from 1997. Grandiose as this may sound, it falls short in two respects: first, modern physics is well known to need the refinement of classical field theory to "quantum field theory" or "high energy physics" and, second, modern foundational mathematics is well known to benefit from the refinement of topos theory to higher topos theory. In the talk I will indicate how there is a natural and useful synthetic axiomatization of high energy physics in higher topos theory.

Related material at the nLab: notes for the talk and background material on synthetic QFT

Conditional expectation as a functor
Vincent Danos (University of Edinburgh)
We present conditional expectation as a functor between the category of probabilistic triples and a category of complete normed cones (after Selinger). The construction relies on an involutive duality between L_infinity and L_1 (which holds for cones, thanks to order continuity conditions; but not for the usual Banach case, where the L_1 double dual is larger than L_1). Using this duality, the construction is a series of easy steps. We show an application to defining approximants for discrete-time Markov chains.

(Joint work with Chaput, Panangaden, and Plotkin.)

Quantum computing in (almost) any category: an introduction to the ZX-calculus
Ross Duncan (University of Strathclyde)
The textbook presentation of quantum mechanics relies of Hilbert spaces, however the phenomena exploited by quantum computers rely on a relatively small number of algebraic structures which are found in many other categories. I'll discuss how to formalise quantum complementarity in any monoidal category and present the ZX-calculus, a beautiful graphical calculus based on this formalisation.

Program semantics, according to Heisenberg and to Schroedinger
Bart Jacobs (Radboud University Nijmegen)
Dualities occur naturally in program semantics, via the duality between state transformer and predicate transformer semantics. In the foundations of quantum mechanics there is a fundamental duality between states and effects; it is commonly associated with differences in approaches of Schroedinger and Heisenberg. The talk presents ongoing work towards a common (categorical) setting in which to describe such program semantics and the associated dualities.

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