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The category of a partitioned fan

Maximilian Kaipel

TL;DR

The paper develops a geometric and categorical framework around partitioned fans to generalize τ-cluster morphism categories. It defines admissible partitions to form the category of a partitioned fan, proves these categories are cubical, and analyzes their classifying spaces as cube complexes with connections to picture groups. Under broad conditions—especially for rank-2 fans and many hyperplane arrangements—the authors establish faithful functors to picture groups and provide criteria ensuring the classifying spaces are K(π,1) spaces. By linking g-vector fans of finite-dimensional algebras to hyperplane arrangements and torsion theory, the work yields new infinite families of algebras whose τ-cluster morphism categories admit faithful functors and K(π,1) classifying spaces, and offers a novel algebraic proof of the relation between an algebra and its g-vector fan. Overall, the results forge a deep connection between combinatorial fan structures, categorical constructions, and stable homotopy properties relevant to τ-tilting theory and representation theory.

Abstract

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the $τ$-cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the $τ$-cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are $K(π,1)$ spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in $\mathbb{R}^2$ and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the $τ$-cluster morphism category admits a faithful functor and for which the cube complexes are $K(π,1)$ spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its $g$-vector fan.

The category of a partitioned fan

TL;DR

The paper develops a geometric and categorical framework around partitioned fans to generalize τ-cluster morphism categories. It defines admissible partitions to form the category of a partitioned fan, proves these categories are cubical, and analyzes their classifying spaces as cube complexes with connections to picture groups. Under broad conditions—especially for rank-2 fans and many hyperplane arrangements—the authors establish faithful functors to picture groups and provide criteria ensuring the classifying spaces are K(π,1) spaces. By linking g-vector fans of finite-dimensional algebras to hyperplane arrangements and torsion theory, the work yields new infinite families of algebras whose τ-cluster morphism categories admit faithful functors and K(π,1) classifying spaces, and offers a novel algebraic proof of the relation between an algebra and its g-vector fan. Overall, the results forge a deep connection between combinatorial fan structures, categorical constructions, and stable homotopy properties relevant to τ-tilting theory and representation theory.

Abstract

In this paper, we introduce the notion of an admissible partition of a simplicial polyhedral fan and define the category of a partitioned fan as a generalisation of the -cluster morphism category of a finite-dimensional algebra. This establishes a complete lattice of categories around the -cluster morphism category, which is closely tied to the fan structure. We prove that the classifying spaces of these categories are cube complexes, which reduces the process of determining if they are spaces to three sufficient conditions. We characterise when these conditions are satisfied for fans in and prove that the first one, the existence of a certain faithful functor, is satisfied for hyperplane arrangements whose normal vectors lie in the positive orthant. As a consequence we obtain a new infinite class of algebras for which the -cluster morphism category admits a faithful functor and for which the cube complexes are spaces. In the final section we also offer a new algebraic proof of the relationship between an algebra and its -vector fan.
Paper Structure (14 sections, 56 theorems, 49 equations, 5 figures)

This paper contains 14 sections, 56 theorems, 49 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on fans and simplicial complexes
  3. The category of a partitioned fan
  4. Cubical categories
  5. The classifying space and picture group
  6. Description of the CW-complex
  7. The picture group of a fan
  8. Eilenberg-MacLane spaces of rank 2
  9. Hyperplane arrangements
  10. Lattice of admissible partitions
  11. Applications to the $\tau$-cluster morphism category
  12. $\tau$-tilting theory
  13. The fan structure of $g$-vectors
  14. $\tau$-cluster morphism category

Key Result

Theorem 1.1

(thm:cubicalcat) Let $\Sigma$ be a simplicial fan and $\mathfrak{P}$ an admissible partition. Then the category $\mathfrak{C}(\Sigma, \mathfrak{P})$ of the partitioned fan is a cubical category and its classifying space $\mathcal{B} \mathfrak{C}(\Sigma, \mathfrak{P})$ thus a cube complex.

Figures (5)

  • Figure 1: A fan and an associated category
  • Figure 2: An example of the simplicial complexes of a fan.
  • Figure 3: The construction of $e([0])$ from factorisation cubes.
  • Figure 4: The classifying space $\mathcal{B} [0] \setminus \mathfrak{C}(\Sigma, \mathfrak{P})$.
  • Figure 5: Last factors and fan poset of fans in $\mathbb{R}^2$.

Theorems & Definitions (120)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 110 more