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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
@article{TAC_2016_31_a0,
author = {Kazunori Noguchi and Kohei Tanaka},
title = {The {Euler} characteristic of an enriched category},
journal = {Theory and applications of categories},
pages = {1--30},
publisher = {mathdoc},
volume = {31},
year = {2016},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2016_31_a0/}
}
Kazunori Noguchi; Kohei Tanaka. The Euler characteristic of an enriched category. Theory and applications of categories, Tome 31 (2016), pp. 1-30. http://geodesic.mathdoc.fr/item/TAC_2016_31_a0/