Title:

  Categorification of Hopf algebras of rooted trees

  Abstract:

  I will exhibit a monoidal structure on the category of finite sets
  indexed by $P$-trees, for a finitary polynomial endofunctor $P$.  This
  structure categorifies the monoid scheme (here meaning `functor from
  semirings to monoids') represented by (a $P$-version of) the
  Connes-Kreimer bialgebra from renormalisation theory.  (The antipode
  arises only after base change from $\N$ to $\Z$.)  The multiplication
  law is itself a polynomial functor, represented by three easily
  described set maps, occurring also in the polynomial representation of
  the free monad on $P$.  (Various related Hopf algebras that have
  appeared in the literature result from varying $P$.)

  The construction itself is not difficult, so most of the talk will be
  spent introducing the involved notions: after some comments about
  combinatorial Hopf algebras and the Connes-Kreimer Hopf algebra in
  particular, I will recall some notions from the theory of polynomial
  functors.  These enter in two different ways: one is the `operad' sort
  of way (where the natural transformations are cartesian), employed
  for talking about trees.  The other is the `categorification-of-
  polynomial-algebra' aspect (where natural transformations are not
  required to be cartesian): the distributive category of polynomial
  functors (in tree-many variables) is the `coordinate ring' of the
  `affine space' given by tree-indexed finite sets.