Let be the groupoid of finite sets and bijections – the core of FinSet. Its groupoid cardinality is the Euler number
Let be a finite crossed complex, (i.e., an omega-groupoid; see the work of Brown and Higgins) such that for any object of the cardinality of the set of -cells with source is independent of the vertex . Then the groupoid cardinality of can be calculated as , much like a usual Euler characteristic. For the case when is a totally free crossed complex, this gives a very neat formula for the groupoid cardinality of the internal hom , in the category of omega-groupoids. Therefore the groupoid cardinality of the function spaces (represented themselves by internal homs) can easily be dealt with if the underlying target space is represented by a omega-groupoid, i.e., has trivial Whitehead products. (This is explored in the papers by Faria Martins and Porter mentioned in the reference list, below.)
Kazunori Noguchi, The Euler characteristic of infinite acyclic categories with filtrations, arxiv/1004.2547
Frank Quinn, 1995, Lectures on axiomatic topological quantum field theory , in D. Freed and K. Uhlenbeck, eds., Geometry and Quantum Field Theory , volume 1 of IAS/Park City Mathematics Series , AMS/IAS.
Revised on November 9, 2012 14:42:10
by Urs Schreiber