论文标题
单宽宽度
Monoidal Width
论文作者
论文摘要
我们引入单体宽度，以衡量单体类别中形态的复杂性。受图形的众所周知的结构宽度度量的启发，例如树的宽度和等级宽度，单体宽度基于句法分解的概念：形态学的单体分解是单体类别语言中的一种表达，其中操作是单体产品和成分，它指定了这种形态学。单体宽度会沿``大''物体沿``大型''对象的组成操作进行了惩罚，同时它鼓励使用单体产品。我们表明，通过选择正确的分解代数进行分解图，我们可以捕获树宽度和等级宽度。对于矩阵，单宽宽度与等级有关。这些示例表明，单宽宽度是在单体类别中建模为形态的过程的结构复杂性的一个很好的衡量。
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.
