We introduce a new type of categorical object called a hom–tensor category and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of hom-braided category and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the Hom–Yang–Baxter equation fits into this framework and how the category of Yetter–Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom–tensor category (respectively a hom-braided category).

中文翻译：

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Hom-Tensor 范畴和 Hom-Yang-Baxter 方程

我们引入了一种称为 hom-tensor 类别的新型分类对象，并表明它为任意 hom-双代数上的模块提供了适当的设置。接下来，我们介绍 hom-braided 类别的概念，并表明这是拟三角 hom-双代数上模块的正确设置。我们还展示了 Hom-Yang-Baxter 方程如何适合这个框架，以及如何将具有双射结构映射的 hom-双代数上的 Yetter-Drinfeld 模块的类别组织为 hom-braided 类别。最后我们证明，在一定条件下，可以从一个 hom-张量范畴（分别是一个 hom-braided 范畴）得到一个张量范畴（分别是一个编织张量范畴）。