This is the last 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 third part relative simplicial monoids are analyzed. Their Moore length is introduced and the equivalence is proven between relative simplicial monoids of Moore length 1, and relative categories of monoids in Part I. This equivalence is obtained in one direction by truncating a simplicial monoid at the first two degrees; and in the other direction by taking the simplicial nerve of a relative category.

中文翻译：

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幺半群的交叉模块 III。摩尔长度 1 的单纯幺半群

这是三篇密切相关的系列论文的最后一部分，其中研究了三个等效结构：幺半群范畴中的内部范畴；根据相对于所选跨度类别的回拉定义 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）。第三部分分析了相对单纯幺半群。介绍了它们的摩尔长度，并证明了摩尔长度为 1 的相对单纯幺半群与第一部分中幺半群的相对类别之间的等价性。这种等价是通过在前两度处截断一个单纯幺半群而在一个方向上获得的；并在另一个方向通过采用相对类别的单纯神经。