Hereditary extriangulated categories: Silting objects, mutation, negative extensions

TL;DR

This paper introduces the theory of hereditary extriangulated categories, connecting mutation theories in representation theory with new categorical structures, and explores negative extensions within these categories.

Contribution

It develops a mutation theory for hereditary extriangulated categories, unifying various mutation concepts like silting, cluster tilting, and tau-tilting mutations.

Findings

Established a mutation theory for maximal rigid objects in hereditary extriangulated categories.

Unified mutation concepts across silting, cluster tilting, and tau-tilting theories.

Provided conditions for the existence of universal balanced negative extensions.

Abstract

In this article, we initiate the study of hereditary extriangulated categories. Many important categories arising in representation theory in connection with various theories of mutation are hereditary extriangulated. Special cases include homotopy categories of 2-term complexes with projective components, which are related to silting mutation, and cluster categories (with relevant relative extriangulated structures) where cluster tilting mutation take place. We prove that there is a theory of irreducible mutation for maximal rigid objects and subcategories in hereditary extriangulated categories of dominant dimension 1. Applied to the examples above, this recovers 2-term silting mutation in triangulated categories and cluster tilting mutation. By constructing suitable extriangulated categories, we also recover tau-tilting mutation for gentle algebras and flips for their non-kissing facets. Combined with results by Adachi-Tsukamoto and Pauksztello-Zvonareva, our mutation also provides mutation for intermediate co-t-structures. In spirit of our earlier work, we study negative extensions in hereditary extriangulated categories. We give sufficient conditions for the existence of universal balanced negative extensions. We explicitly compute certain, a priori non-universal, versions of negative extensions in hereditary categories constructed from triangulated categories with rigid subcategories. We discuss examples where these two constructions give the same delta-functors and where they disagree.