**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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