Groupoids and polynomial functors in the 
combinatorics of Quantum Field Theory

Groupoids generalise both sets and groups, and are very useful
to deal with combinatorial problems involving symmetries.  The
main point of this talk is the insight that sums weighted by
symmetry factors can be seen as relative homotopy cardinalities
of groupoids, and that manipulation with such series can be
interpreted as constructions with groupoids.  This viewpoint
reveals some notions in the combinatorics of Quantum Field
Theory to be special cases of more general constructions of
wider interest.  For example, (combinatorial) Green functions as
solutions to (combinatorial) Dyson-Schwinger equations can be
seen as an instance of the general fact that if P is a finitary
polynomial endofunctor over groupoids, then the groupoid of
P-trees is the homotopy least-fixpoint of 1+P (a result
originating in theoretical computer science, where it is the
standard approach to inductive data types).  Rather than going
into technical details, I hope to convey the overall ideas by
starting gently with some basic facts about groupoids and
homotopy cardinality, then introduce polynomial functors and
explain their relationship with trees, and finally give examples
of polynomial endofunctors P and their corresponding P-trees
motivated by QFT.