Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.

中文翻译：

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Gabriel-Zisman 上同调和光谱序列

扩展 Gabriel 和 Zisman 的构造，我们为单纯集的上同调和同调开发了一个函子框架，其中具有由单纯形范畴上的函子到阿贝尔范畴中给出的非常一般的系数系统。此外，我们为任何单纯集映射构造 Leray 型谱序列。我们还表明，这些构造概括和统一了小类别的上同调和同源性的各种现有版本，并且作为奖励提供了对其功能性的新见解。