Euler characteristic for weak n-categories and $\left(\infty,1\right)$-categories

TL;DR

This paper extends the concept of Euler characteristic from finite strict categories to certain finite weak 2-categories, weak n-categories, and some finite (,1)-categories, broadening its applicability.

Contribution

It introduces a new extension of Euler characteristic to weak 2-categories and sketches its extension to weak n-categories and (,1)-categories, building on Leinster's work.

Findings

Extended Euler characteristic to weak 2-categories.

Outlined potential extension to weak n-categories.

Discussed applicability to (,1)-categories.

Abstract

The Euler characteristic was defined for finite strict n-categories by Leinster using the theory of enriched categories. This was an extension of some of his earlier work, which defined Euler characteristic for finite categories. Building on Leinster's work, we extend the notion of Euler characteristic to certain finite weak 2-categories and present a sketch of a similar extension to weak n-categories. We also discuss the extension of the Euler characteristic to certain finite $\left(\infty,1\right)$-categories.