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度的度量。受树宽、秩宽等图论中经典结构复杂度度量的启发，幺半宽度基于句法分解的概念：态射的幺半分解是幺半范畴语言中的表达式（其中运算包括幺半积与复合），该表达式唯一确定该态射。幺半宽度惩罚沿"大"对象的复合运算，同时鼓励使用幺半积。我们证明，通过选取合适的图分解范畴代数结构，可以捕捉树宽与秩宽。对于矩阵而言，幺半宽度与矩阵秩相关。这些示例表明，幺半宽度可作为以幺半范畴态射建模的过程结构复杂度的有效度量。