### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- Enriched categories
- Euler characteristic
- Monoidal model categories

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*Theory and Applications of Categories*,

*31*, 1-30.

**The euler characteristic of an enriched category.** / Noguchi, Kazunori; Tanaka, Kohei.

Research output: Contribution to journal › Article

*Theory and Applications of Categories*, vol. 31, pp. 1-30.

}

TY - JOUR

T1 - The euler characteristic of an enriched category

AU - Noguchi, Kazunori

AU - Tanaka, Kohei

PY - 2016/1/3

Y1 - 2016/1/3

N2 - 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).

AB - 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).

KW - Enriched categories

KW - Euler characteristic

KW - Monoidal model categories

UR - http://www.scopus.com/inward/record.url?scp=84955578915&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84955578915&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84955578915

VL - 31

SP - 1

EP - 30

JO - Theory and Applications of Categories

JF - Theory and Applications of Categories

SN - 1201-561X

ER -