This is the first 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 first part the theory of relative pullbacks is worked out leading to the definition of a relative category.

中文翻译：

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幺半群的交叉模 I：相对类别

这是三篇密切相关的论文系列的第一部分，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 - 相对于此类跨度的幺半群的交叉模 - 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）。在第一部分中，研究了相对回调理论，从而得出了相对类别的定义。