Title:
  Polynomial functors, negative sets, and moduli of punctured Riemann
  spheres.

Abstract:
  A polynomial functor (of one variable) is an endofunctor on the category
  of sets, built from disjoint unions, products, and exponentiation.  They
  behave a lot like polynomials over the natural numbers: you can add and
  multiply them or substitute them into each other; you can also
  differentiate them, and there is a Leibniz rule and a chain rule, as I
  shall explain.  Now, in order to model integer coefficients (and
  exponents) a notion of negative sets is required.  I'll introduce
  Schanuel's category of polyhedral sets, for which a certain Euler measure
  provides a generalisation of cardinality that can take negative integer
  values.  To finish with an interesting example, I'll describe the
  polynomial functors associated to the moduli spaces of Riemann spheres
  with $3+n$ punctures.  The puncture operator corresponds precisely to
  differentiation, and in the setting of polyhedral sets, these functors are 
  models for the familiar Laurent polynomials $-n!(-X)^{-n-1} = D^n X^{-1}$.