Three lectures on polynomial functors
In the first lecture, I'll develop some elementary aspects of
the theory of polynomial functors. The main result will be
Tambara's theorem, stating that the category of finite
polynomial functors is the Lawvere theory for commutative
semirings. I will also explain one interpretation of polynomial
functors in type theory: their initial algebras are inductive
data types.
In the second lecture, the main themes will be trees and
polynomial monads. While specific kinds of trees (P-trees) can
be viewed as forming the initial algebra of a polynomial
endofunctor 1+P, it is also possible to define trees more
combinatorially: a tree is itself a certain polynomial
endofunctor. This leads to the interpretation of P-trees as
operations of the free monad on P, emphasising another aspect of
trees, namely substitution. Polynomial monads are essentially
operads, and the polynomial viewpoint turns out to be useful
in handling the combinatorics underlying algebraic structures,
for example to construct model structures on categories of
algebras.
Finally in the third lecture, I'll talk about combinatorial
species and polynomial functors over groupoids and infinity
groupoids, covering first the necessary background (which is
not so much). Groupoid coefficients are needed to transparently
handle combinatorial structures with symmetries, and in
particular combinatorial structures with so many symmetries that
they don't form classical species (such as Feynman graphs).
Infinity-groupoid coefficients serve to capture higher
homotopical data, such as inherent in intensional type theory.