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：我们引入幺半宽度作为幺半范畴中态射复杂性的度量。受图论中树宽、秩宽等经典结构宽度度量的启发，幺半宽度基于句法分解的概念：态射的幺半分解是用幺半范畴语言表示的表达式，其中运算包括幺半乘积与复合，该表达式唯一确定该态射。幺半宽度对涉及"大"对象的复合运算施加惩罚，同时鼓励使用幺半乘积。研究表明，通过选择合适的图分解范畴代数，可以刻画树宽与秩宽。对于矩阵而言，幺半宽度与秩相关。这些实例表明，幺半宽度可作为建模为幺半范畴中态射的过程的结构复杂性的有效度量。