With a view on applications in computing, in particular concurrency theory and higher-dimensional rewriting, we develop notions of $n$-fold monoid and comonoid objects in $n$-fold monoidal categories and bicategories. We present a series of examples for these structures from various domains, including a categorical model for a communication protocol and a lax $n$-fold relational monoid, which has previously been used implicitly for higher-dimensional rewriting and which specialises in a natural way to strict $n$-categories. A special set of examples is built around modules and algebras of the boolean semiring, which allows us to deal with semilattices, additively idempotent semirings and quantales using tools from classical algebra.

翻译：着眼于计算领域的应用，特别是并发理论与高维重写，我们发展了n-重幺半范畴与双范畴中n-重幺半对象与余幺半对象的概念。我们为这些结构提供了一系列来自不同领域的实例，包括通信协议的范畴模型、以及先前已隐式用于高维重写的松弛n-重关系幺半群——该结构能以自然方式特化为严格n-范畴。特别地，我们围绕布尔半环的模与代数构建了一组示例，这使得我们能够运用经典代数的工具处理半格、加法幂等半环与量格。