Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner’s bigraphs and crossed modules are considered.

中文翻译：

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（弱）单调类别的发生率双代数

研究了 Joni 和 Rota 意义上的类别的发生余代数，特别是类别上的幺半积将这些变成（弱）双代数的情况。讨论了与组合 Hopf 代数理论和 Hopf quivers 理论的重叠，并考虑了包括树、倾斜形状、米尔纳二合图和交叉模块的例子。