In this short note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by a finite category, but the behavior is more uniform when we index by finite quasicategories instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.

中文翻译：

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有限同伦共边的欧拉特征

在这个简短的说明中，我们提供了有限元胞复合体的有限同伦共限的欧拉特征的计算，它仅依赖于每个空间的欧拉特征并且类似于莫比乌斯反演。当 colimit 由有限类别索引时，结果的版本是已知的，但当我们改为由有限准类别索引时，行为更加统一。该公式同时推广了同伦推出的欧拉特性的加法公式和纤维丛欧拉特性的乘法公式。