Let $M$ be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in $M$. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normalized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when $M$ is the category of simplicial sets or the category of chain complexes over a commutative ring.
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机译:假设$ M $是具有等价类的弱等价类以及具有适当相容性的类集和类同义类的分类束的单等分类别。我们定义并研究了以$ M $表示的单边体图的正态性和共形体图的正态性的同态不变概念。这些概念概括了主束和交叉模块,并被足够好的单曲面函子(例如归一化链复杂函子)保留。当$ M $是简单集的类别或交换环上的链络合物的类别时,我们提供同型正态图和同型正态图的示例的几个显式类。
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