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.