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$τ$-cluster morphism categories of factor algebras

Maximilian Kaipel

TL;DR

The paper develops a lattice-theoretic construction of the τ-cluster morphism category $\mathfrak{T}(A)$ from the lattice of torsion classes $\mathrm{tors}\,A$ and shows that, when $\mathrm{tors}\,A$ is finite, the category is determined up to equivalence by the underlying lattice. It introduces a functor $F_I:\mathfrak{T}(A)\to\mathfrak{T}(A/I)$ induced by the lattice congruence from an ideal $I$, and characterizes when $F_I$ is surjective on objects, faithful, and full; in particular, fullness occurs precisely when $\mathrm{tors}\,A\cong\mathrm{tors}\,A/I\times\mathrm{tors}\,B$ for some finite-dimensional $B$, while faithfulness requires $\mathrm{tors}\,A\cong\mathrm{tors}\,A/I$. The authors further relate classifying spaces and picture groups under quotienting, and show that every morphism of $\mathfrak{T}(A/I)$ lies in the image of $F_I$, with $F_I$ being a regular epimorphism in the category of small categories. They demonstrate lifting of $\tau$-perpendicular intervals and provide examples to illustrate the necessity of finiteness, and discuss connections with the previously defined $\mathfrak{W}(A)$ and $\mathfrak{C}(A)$, as well as implications for brick-labellings, $g$-vector fans, and maximal green sequences. Overall, the work reveals that the τ-cluster morphism category is governed by lattice structure, enabling a combinatorial route to understanding quotient algebras and related topological and group-theoretic invariants in this part of representation theory.

Abstract

We take a novel lattice-theoretic approach to the $τ$-cluster morphism category $\mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $\mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: \mathfrak{T}(A) \to \mathfrak{T}(A/I)$. If $\mathrm{tors } A$ is finite, $F_I$ is a regular epimorphism in the category of small categories and we characterise when $F_I$ is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the $τ$-cluster morphism category up to equivalence.

$τ$-cluster morphism categories of factor algebras

TL;DR

The paper develops a lattice-theoretic construction of the τ-cluster morphism category from the lattice of torsion classes and shows that, when is finite, the category is determined up to equivalence by the underlying lattice. It introduces a functor induced by the lattice congruence from an ideal , and characterizes when is surjective on objects, faithful, and full; in particular, fullness occurs precisely when for some finite-dimensional , while faithfulness requires . The authors further relate classifying spaces and picture groups under quotienting, and show that every morphism of lies in the image of , with being a regular epimorphism in the category of small categories. They demonstrate lifting of -perpendicular intervals and provide examples to illustrate the necessity of finiteness, and discuss connections with the previously defined and , as well as implications for brick-labellings, -vector fans, and maximal green sequences. Overall, the work reveals that the τ-cluster morphism category is governed by lattice structure, enabling a combinatorial route to understanding quotient algebras and related topological and group-theoretic invariants in this part of representation theory.

Abstract

We take a novel lattice-theoretic approach to the -cluster morphism category of a finite-dimensional algebra and define the category via the lattice of torsion classes . Using the lattice congruence induced by an ideal of we establish a functor . If is finite, is a regular epimorphism in the category of small categories and we characterise when is full and faithful. The construction is purely combinatorial, meaning that the lattice of torsion classes determines the -cluster morphism category up to equivalence.
Paper Structure (10 sections, 41 theorems, 72 equations, 3 tables)

This paper contains 10 sections, 41 theorems, 72 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. $\tau$-tilting theory
  4. Geometric definition
  5. Definition from the lattice of torsion classes
  6. Invariance under $\tau$-tilting equivalence
  7. Factor algebras and lattice congruences
  8. Epimorphisms and lifting $\tau$-perpendicular intervals
  9. Classifying spaces and picture groups
  10. Examples

Key Result

Theorem 1.1

Let $A$ be a finite-dimensional algebra, then we may define the $\tau$-cluster morphism category $\mathfrak{T}(A)$ from the lattice of torsion classes $\mathop{\mathrm{tors}}\nolimits A$. Moreover, if $\mathop{\mathrm{tors}}\nolimits A$ is finite, the category is determined entirely by the underlyin

Theorems & Definitions (98)

  • Theorem 1.1
  • proof
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Example 2.4
  • ...and 88 more