Sonlu kategorilerin Euler ölçüsü

We associate a rational number χ(A) to every category A whose object and morphism sets are finite. The assignment χ is additive under disjoint union and it preserves products. Leinster's Euler characteristic χLein and χ agrees whenever χLein is defined. Hence χ is different from the series Euler characteristic χΣ and χ is preserved under the weak equivalences of canonical model structure when it is restricted to the family of categories for which χLein is defined. However, χ is not preserved under the weak equivalences of canonical model structure on its whole domain. For this reason χ is not called the Euler characteristic. When the domain of χ is restricted to the family of categories admitting a weighting, χ satisfies the inclusion exclusion principle. Hence we can call this restriction the Euler measure. By abuse of notation we will denote this restriction by χ again. Since the family of categories admitting both weighting and coweighting is contained by the family of categories admitting weighting, the Euler measure of categories is a proper extension of Leinster's Euler characteristic. We also showed that Leinster's formula for the Grothendieck construction is still valid for diagrams from a poset to the categories in the domain of this Euler measure. The situation for the Thomason model structure is more intricate. We give an example to show that none of χ, χLein and χΣ is invariant under the weak equivalences of the Thomason model structure and show that such examples can be eliminated by putting extra conditions on weak equivalences of the Thomason model structure.