### Abstract

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).

Original language | English |
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Pages (from-to) | 1-30 |

Number of pages | 30 |

Journal | Theory and Applications of Categories |

Volume | 31 |

Publication status | Published - 2016 Jan 3 |

Externally published | Yes |

### Keywords

- Enriched categories
- Euler characteristic
- Monoidal model categories

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

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## Cite this

*Theory and Applications of Categories*,

*31*, 1-30.