Polynomial functors over groupoids:
from program semantics to quantum field theory -- and back
The theory of polynomial functors has origins in program semantics:
on one hand polynomial functors serve as data type constructors, often
called containers, and on the other hand their initial algebras are
well-founded trees, accounting for inductive data types. It is useful
to upgrade the theory to groupoid coefficients instead of set
coefficients: this allows to capture also data types with symmetries,
such as multisets, and gives a more flexible notion of well-founded tree,
covering in particular non-planar trees. Polynomial functors over
groupoids cover also combinatorial species as well as Baez-Dolan stuff
types, and allow for ``type-theoretic'' methods in combinatorics.
More specifically, Feynman graphs, as used in pertubative quantum field
theory, can be interpreted as well-founded trees. The fixpoint equations
that generate them are the so-called combinatorial Dyson-Schwinger
equations, a combinatorial skeleton of the quantum equations of motion.
They are of great interest in quantum field theory, and categorical
methods can help understanding them. Conversely, methods used in
combinatorics and QFT, such as Hopf algebras and renormalisation schemes,
should have applications in computer science. Manin has recently
suggested a renormalisation approach to the theory of computation, and
to the halting problem in particular. Polynomial functors should be a
convenient framework to understand these connections.
Some of these connections are a bit speculative at the moment, and I
am not an expert in either program semantics or quantum field theory.
The goal of the lectures is first of all to explain the parts that are
on reasonably firm footing, and second to try to involve theoretical
computer scientists to help with the developments. No prior knowledge
of physics is assumed, the main prerequisite being familiarity with
category theory.