Algebra and Geometry,
Université Catholique de
Belgium, 27 May 2011.
Abstract Every small category A has a classifying space BA, whose homotopy type depends not only on the underlying graph of A, but also on its composition and identities. However, the Euler characteristic of BA depends only on the underlying graph. This talk can be understood as an exploration of this fact.
The key idea is Möbius inversion. The original, number-theoretic, Möbius inversion takes place in the poset of positive integers ordered by divisibility. It has been generalized to categories in two different ways. The first, which I will call fine Möbius inversion, was introduced independently by Pierre Leroux and John Haigh. It does depend on the composition of the category in which it takes place. The second, coarse Möbius inversion, was introduced independently by Haigh and the speaker. It does not depend on the composition. I will explain what fine and coarse Möbius inversion are, and how they are related to each other and to Euler characteristic. The aim is to shed light on the fact that the Euler characteristic of a category is independent of its composition.
Slides In this pdf file
Discussion Möbius inversion for categories