We introduce monoidal width as a measure of complexity for morphisms in monoidal categories. Inspired by well-known structural width measures for graphs, like tree width and rank width, monoidal width is based on a notion of syntactic decomposition: a monoidal decomposition of a morphism is an expression in the language of monoidal categories, where operations are monoidal products and compositions, that specifies this morphism. Monoidal width penalises the composition operation along ``big'' objects, while it encourages the use of monoidal products. We show that, by choosing the correct categorical algebra for decomposing graphs, we can capture tree width and rank width. For matrices, monoidal width is related to the rank. These examples suggest monoidal width as a good measure for structural complexity of processes modelled as morphisms in monoidal categories.

翻译：我们提出幺半宽度（monoidal width）作为幺半范畴中态射复杂度的度量方法。受图论中树宽和秩宽等经典结构宽度度量的启发，幺半宽度基于语法分解的概念：态射的幺半分解是幺半范畴语言中的一个表达式，其中运算包括幺半积与复合，用于刻画该态射。幺半宽度惩罚沿“大”对象进行的复合运算，同时鼓励使用幺半积。我们证明，通过选择正确的图分解范畴代数，可以刻画树宽和秩宽。对矩阵而言，幺半宽度与秩相关。这些例子表明，幺半宽度可作为建模为幺半范畴中态射的过程结构复杂性的有效度量。