Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology

Luigi Caputi, Carlo Collari, Eric Ramos

Abstract

Building upon previous works of Proudfoot and Ramos, and using the categorical framework of Sam and Snowden, we extend the weak categorical minor theorem from undirected graphs to quivers. As case of study, we investigate the consequences on the homology of multipath complexes; eg. on its torsion. Further, we prove a comparison result: we show that, when restricted to directed graphs without oriented cycles, multipath complexes and matching complexes yield functors which commute up to a blow-up operation on directed graphs. We use this fact to compute the homotopy type of matching complexes for a certain class of bipartite graphs also known as half-graphs or ladders. We complement the work with a study of the (representation) category of cones, and with analysing related consequences on magnitude cohomology of quivers.

The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology

Abstract

Building upon previous works of Proudfoot and Ramos, and using the categorical framework of Sam and Snowden, we extend the weak categorical minor theorem from undirected graphs to quivers. As case of study, we investigate the consequences on the homology of multipath complexes; eg. on its torsion. Further, we prove a comparison result: we show that, when restricted to directed graphs without oriented cycles, multipath complexes and matching complexes yield functors which commute up to a blow-up operation on directed graphs. We use this fact to compute the homotopy type of matching complexes for a certain class of bipartite graphs also known as half-graphs or ladders. We complement the work with a study of the (representation) category of cones, and with analysing related consequences on magnitude cohomology of quivers.
Paper Structure (10 sections, 32 theorems, 21 equations, 6 figures)

This paper contains 10 sections, 32 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Basic notions
  3. Quivers
  4. Finitely generated $\mathbf{C}$-modules
  5. Gröbner properties of $\mathbf{Quiver}$ and $\mathbf{Digraph}$
  6. Categories of directed cones
  7. Applications to multipath complexes
  8. Multipath complexes and their torsion
  9. Multipath complexes and matching complexes
  10. Applications to magnitude homology

Key Result

Theorem 1.1

The category $\mathbf{Quiver}_{\leq g}^\mathrm{op}$ is quasi-Gröbner.

Figures (6)

  • Figure 1: The bipartite graph ${\tt B}_3$.
  • Figure 2: The alternating linear quiver ${\tt A}_n$ on $n+1$ vertices. The edge between $v_{n -1}$ and $v_{n}$ can be oriented either way depending on the parity of $n$.
  • Figure 3: The graph ${\tt S}(m_1,m_2,m_2', m_3)$. The edge in blue goes from $v_1$ to $v_2$.
  • Figure 4: The coherently oriented linear graph ${\tt I}_3$ (top left), the multipath complex $X({\tt I}_3)$ (top right), and the path poset $P({\tt I}_3)$ (bottom).
  • Figure 5: The graph ${\tt D}_{3,2}$.
  • ...and 1 more figures

Theorems & Definitions (82)

  • Theorem 1.1: Weak categorical quiver minor theorem
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6: FI_Noetherian
  • Definition 2.7
  • ...and 72 more