This is the second part of a series of three strongly related papers in which three equivalent structures are studied: internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans crossed modules of monoids relative to this class of spans simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this second part we define relative crossed modules of monoids and prove their equivalence with the relative categories of Part

中文翻译：

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Monoids 的交叉模块 II：相对交叉模块

这是三篇密切相关的系列论文的第二部分，其中研究了三个等效结构：幺半群范畴中的内部范畴；定义为相对于所选跨度类别的回拉与相对于此类跨度的幺半群跨模数相对于此类跨度的所谓摩尔长度为 1 的单纯幺半群。所涵盖的幺半群最重要的例子是小类别（在跨度类别中被视为幺半群）和对称幺半群类别中的双幺半群（在 comonoids 类别中被视为幺半群）。在第二部分中，我们定义了幺半群的相对交叉模块，并证明了它们与 Part 的相对类别的等价性