Monoidal Width

TL;DR

The paper introduces monoidal width, a new measure of complexity for morphisms in monoidal categories, capturing structural properties similar to graph width measures and relating to matrix rank.

Contribution

It defines monoidal width based on monoidal decompositions, connecting it to existing graph measures and matrix rank, providing a unified complexity measure.

Findings

Monoidal width generalizes tree width and rank width.

For matrices, monoidal width correlates with matrix rank.

It offers a new perspective on structural complexity in categorical models.

Abstract

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.