We develop the homotopy theory of Euler characteristic (magnitude) of a category enriched in a monoidal model category. If a monoidal model category $V$ is equipped with an Euler characteristic that is compatible with weak equivalences and fibrations in $V$, then our Euler characteristic of $V$-enriched categories is also compatible with weak equivalences and fibrations in the canonical model structure on the category of $V$-enriched categories. In particular, we focus on the case of topological categories; i.e., categories enriched in the category of topological spaces. As its application, we obtain the ordinary Euler characteristic of a cellular stratified space $X$ by computing the Euler characteristic of the face category $C(X)$.
Keywords: Euler characteristic, enriched categories, monoidal model categories
2010 MSC: 18D20; 55U35
Theory and Applications of Categories, Vol. 31, 2016, No. 1, pp 1-30.