We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.

中文翻译：

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分组直至一致性关系，从分类组到C交叉模块

我们介绍了c-group的概念，它是由等价关系决定的一个组，并考虑相应的类别。扩展，操作和交叉模块（c交叉模块）在此类别中定义，并且构造半直接乘积。我们证明每个分类组都产生一个c-group和一个c-crossed模块，这是我们定义的连通，特殊且严格的c-crossed模块。此处获得的结果将用于证明类别组类别以及连接的，特殊的和严格的c交叉模块的等价性。