Alex Nash: Reimagining the Euler Characteristic: from Polyhedra to Lattices to Categories

Abstract.

We will define the Euler characteristic as a sort of “zero dimensional measure” on the lattice of polyconvex subsets of \(\mathbb{R}^d\). This enables us to view the Euler characteristic as an invariant first and foremost, and then later derive the classical alternating sum formula as a computational tool. This change in perspective facilitates the process of generalizing the Euler characteristic to other mathematical settings. We then show how a similar process may be carried out directly on the lattice of finite abstract simplicial complexes with a fixed vertex set. From that, we can draw on the connection between simplicial complexes and posets to present a definition of the Euler characteristic of a finite poset in terms of its Möbius function. Finally, we show how Euler characteristic can be generalised to a certain class of categories, which includes some categories whose classifying space is not equivalent to a finite complex. In particular, we will see examples where the Euler characteristic is not an integer.