Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monoidal Width

Elena Di Lavore, Paweł Sobociński

TL;DR

Monoidal width proposes a unifying, syntax-driven complexity measure for morphisms in monoidal categories by quantifying the efficiency of monoidal decompositions. By selecting appropriate decomposition algebras, the paper shows that tree width and rank width arise as instances, via cospans of hypergraphs and graphs with boundaries, respectively, while also relating monoidal width to matrix rank. The results provide a principled, algebraic pathway to analyze structural complexity of processes modeled as morphisms, and they point toward potential fixed-parameter tractability results grounded in width. The work advances a program to leverage monoidal categories as a versatile language for describing and analyzing decompositions of graph-like structures and other networked systems.

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.

Monoidal Width

TL;DR

Monoidal width proposes a unifying, syntax-driven complexity measure for morphisms in monoidal categories by quantifying the efficiency of monoidal decompositions. By selecting appropriate decomposition algebras, the paper shows that tree width and rank width arise as instances, via cospans of hypergraphs and graphs with boundaries, respectively, while also relating monoidal width to matrix rank. The results provide a principled, algebraic pathway to analyze structural complexity of processes modeled as morphisms, and they point toward potential fixed-parameter tractability results grounded in width. The work advances a program to leverage monoidal categories as a versatile language for describing and analyzing decompositions of graph-like structures and other networked systems.

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.
Paper Structure (22 sections, 34 theorems, 89 equations, 9 figures)

This paper contains 22 sections, 34 theorems, 89 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Contribution.
  3. String diagrams.
  4. Related work.
  5. Synopsis.
  6. Monoidal width
  7. Monoidal width of copy
  8. A monoidal algebra for tree width
  9. Background: tree width and branch width
  10. Hypergraphs with sources and inductive definition
  11. Cospans of hypergraphs
  12. String diagrams for cospans of hypergraphs
  13. Tree width as monoidal width
  14. Monoidal width in matrices
  15. The prop of matrices
  16. ...and 7 more sections

Key Result

Lemma 2.8

Let $\mathsf{C}$ be a symmetric monoidal category with coherent copying. Suppose that $\mathcal{A}$ contains $\begin{tikzpicture}[baseline=-0.1cm, scale = 0.5] \node(n)[dot={copycolor}{0.35}]at(0,0){}; \coordinate(o1)at(0.5,0.3){}; \coordinate(o2)at(0.5,-0.3){}; \coordinate(i)at(-0.5 Then there is a monoidal decomposition $\mathcal{C}_{\overline{X}}(d)$ of $\gamma_{\overline{X}}(f)

Figures (9)

  • Figure 1: This morphism can be decomposed in two different ways: $(f \otimes f') \mathbin{;} (g \otimes g') = (f \mathbin{;} g) \otimes(f' \mathbin{;} g')$.
  • Figure 2: A tree decomposition cuts the graph along its vertices.
  • Figure 3: A cut and its matrix in a rank decomposition.
  • Figure 4: Sources are marked vertices in the graph and are thought of as an interface that can be glued with that of another graph.
  • Figure 5: Generators and axioms of a special Frobenius monoid.
  • ...and 4 more figures

Theorems & Definitions (104)

  • Definition 2.1: Monoidal decomposition
  • Definition 2.2
  • Definition 2.3: Monoidal width
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • Definition 3.1
  • ...and 94 more