# Maximal $\tau_d$-rigid pairs

## TL;DR

This paper extends the bijection between cluster tilting objects and support $ au$-tilting pairs to higher $(d+2)$-angulated categories, introducing maximal $ au_d$-rigid pairs as a higher analogue.

## Contribution

It establishes a higher analogue of the known bijection in 2-Calabi--Yau categories, using maximal $ au_d$-rigid pairs in $(d+2)$-angulated categories.

## Key findings

- Established a bijection between maximal $ au_d$-rigid pairs and support $ au_d$-tilting pairs.
- Extended known results from 2-Calabi--Yau categories to higher $(d+2)$-angulated categories.
- Introduced the concept of maximal $ au_d$-rigid pairs as a higher analogue.

## Abstract

Let $\mathscr T$ be a $2$-Calabi--Yau triangulated category, $T$ a cluster tilting object with endomorphism algebra $\Gamma$. Consider the functor $\mathscr T( T,- ) : \mathscr T \rightarrow \mod \Gamma$. It induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support $\tau$-tilting pairs. This is due to Adachi, Iyama, and Reiten. The notion of $( d+2 )$-angulated categories is a higher analogue of triangulated categories. We show a higher analogue of the above result, based on the notion of maximal $\tau_d$-rigid pairs.