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Rigid Matroid Categories

Kevin Purbhoo

Abstract

We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples include: deletion and contraction, 2-sum, series and parallel connections, the Tutte polynomial, gammoids, positroids, matroids representable over an infinite field, M-convex sets, and matroids associated to stable polynomials.

Rigid Matroid Categories

Abstract

We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples include: deletion and contraction, 2-sum, series and parallel connections, the Tutte polynomial, gammoids, positroids, matroids representable over an infinite field, M-convex sets, and matroids associated to stable polynomials.
Paper Structure (15 sections, 23 theorems, 56 equations, 7 figures)

This paper contains 15 sections, 23 theorems, 56 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Matroids
  3. Composition
  4. Outline
  5. Matroids as relations
  6. Functorial constructions
  7. Hom-functor and related categories
  8. Deletion, contraction, and symmetrization
  9. Lax composition
  10. The category ${\mathbf{Mtd}^\bullet}$
  11. Deletion and contraction in ${\mathbf{Mtd}^\bullet}$ and related categories
  12. Structure theorem
  13. Enriched matroid categories
  14. Dominant morphisms
  15. Associativity

Key Result

Theorem 2.15

For finite sets $A,B,C$, if $\lambda \in \mathrm{Exch}(2^{A}, 2^{B})$ and $\mu \in \mathrm{Exch}(2^{B}, 2^{C})$ then $\lambda \circ \mu \in \mathrm{Exch}(2^{A},2^{C})$.

Figures (7)

  • Figure 3.1: A directed graph $G \in \mathrm{Mor}_\mathbf{DGraph}(\{1,2,3,4\},\{5,6,7,8,9\})$. $G$ includes the isolated arrow $4 \to 9$.
  • Figure 3.2: Composition in $\mathbf{DGraph}$.
  • Figure 3.3: A bicoloured directed graph $G \in \mathrm{Mor}_\mathbf{BDGraph}(\{1,2,3\},\{4,5,6,7\})$ (left). The orientations of the edges and half-edges of $G$ can be changed to give a perfect orientation (right).
  • Figure 3.4: Converting a directed graph $G$ into a bicoloured directed graph $G^{\bullet\circ}$.
  • Figure 7.1: Schematic diagram of a composition $\Pi_\bullet(\mu_0, \mu_1, \mu_2, \mu_3)$.
  • ...and 2 more figures

Theorems & Definitions (108)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 98 more