---
abstract: 'We give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa–Todorov cluster categories of Dynkin type $A_{ \infty }$.'
address:
- 'Gratz: School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, United Kingdom'
- 'Holm: Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany'
- 'Jørgensen: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom'
author:
- Sira Gratz
- Thorsten Holm
- Peter Jørgensen
title: 'Cluster tilting subcategories and torsion pairs in Igusa–Todorov cluster categories of Dynkin type $A_{ \infty }$'
---

Introduction {#sec:introduction}
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Let $\cC( A_n )$ be the cluster category of Dynkin type $A_n$, see [@BMRRT sec. 1] and [@CCS]. It is well known that $\cC( A_n )$ has a combinatorial model by an $( n+3 )$-gon $P$. The indecomposable objects are in bijection with the diagonals of $P$, and non-vanishing $\Ext^1$ groups correspond to crossing diagonals.

The combinatorial model has two key properties: Cluster tilting subcategories of $\cC( A_n )$ correspond to triangulations of $P$, and torsion pairs in $\cC( A_n )$ correspond to so-called Ptolemy diagrams in $P$. The 