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arXiv 2403.080872024-03-12math.CTmath.LOmath.RA

拓扑斯中的Hochschild上同调

Hochschild cohomology in toposes

Cameron Michie, Ivan Tomasic

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AI总结：

本文在环化拓扑斯中建立内部Hochschild上同调理论，构造内部Hochschild上链复形，特化至差分集合拓扑斯得到内部差分Hochschild上同调，并通过谱序列与绝对上同调比较，系统处理张量积以完善文献。

摘要：

我们在环化拓扑斯中发展了内部Hochschild上同调理论。我们通过内部Hochschild上链复形以及导出函子/拓扑斯上同调理论来构造它，并讨论其与绝对Hochschild上同调的关系。通过特化到差分集合的拓扑斯，我们获得了内部差分Hochschild上同调理论，并通过Grothendieck和超上同调谱序列将其与绝对Hochschild上同调进行比较。我们对合适拓扑斯中的张量积进行了系统而详细的处理，以期完善现有文献。

英文摘要：

We develop a theory of internal Hochschild cohomology in a ringed topos. We construct it via the internal Hochschild cochain complex, as well as through derived functor/topos cohomology theory, and discuss its relationship to the absolute Hochschild cohomology. By specialising to the topos of difference sets, we obtain a theory of internal difference Hochschild cohomology, and compare it to the absolute Hochschild cohomology through the Grothendieck and hypercohomology spectral sequences. We provide a systematic and detailed treatment of tensor products in suitable toposes in hope to complete the existing literature.

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