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The Euler characteristic of an enriched category

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Kazunori Noguchi and Kohei Tanaka

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.

Published 2016-01-03.

http://www.tac.mta.ca/tac/volumes/31/1/31-01.pdf

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