# Indecomposable objects determined by their index in Higher Homological Algebra

## TL;DR

This paper extends the concept that indecomposable objects are uniquely identified by their index from 2-Calabi-Yau triangulated categories to higher-dimensional $(d+2)$-angulated categories, under certain assumptions.

## Contribution

It proves that, with a technical assumption, indecomposable objects in higher $(d+2)$-angulated categories are uniquely determined by their index, generalizing previous results.

## Key findings

- Indecomposable objects are uniquely identified by their index in higher categories.
- The result applies to higher dimensional cluster categories.
- Provides a new understanding of object classification in higher homological algebra.

## Abstract

Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely defined by its index with respect to $\mathscr{T}$. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate an application of this result in higher dimensional cluster categories.