Title: Mšbius inversion in general categories
Abstract: I will report on recent joint work with Imma G‡lvez
and Andy Tonks on Mšbius inversion in general categories (with
coefficients in groupoids). I will start by briefly recalling
the classical theory, from Mšbius to Rota, and then explain the
notion of Mšbius category of Leroux; here subtle finiteness
conditions are required in order to get well-defined sums. The
theory of Mšbius inversion for general categories is achieved by
working with groupoids as coefficients instead of numbers. At
the same time this turns all constructions and proofs
"bijective" instead of algebraic. More precisely, the equations
obtained are equivalences of groupoids, and the techniques have
a homotopy flavour. Extracting algebraic identities is a matter
to taking homotopy cardinality, and for this some finiteness is
obviously required. It is a homotopy finiteness weaker than
Leroux's, and covers examples that are not Mšbius categories in
the sense of Leroux, for example categories of trees. The
Connes-Kreimer Hopf algebra of trees arises directly as the
incidence algebra of the category of trees, without the need of
constructing auxiliary posets as done by DŸr.