Auslander's defects over extriangulated categories: an application for the General Heart Construction
Yasuaki Ogawa
TL;DR
This work generalizes Auslander's formula to extriangulated categories by constructing a localization sequence $\\operatorname{def}\\mathcal{C} \to \\operatorname{mod}\\mathcal{C} \to \\operatorname{lex}\\mathcal{C}$ and analyzing the defect subcategory via the functor $E_\mathcal{C}=Q\circ\\mathbb{Y}$. It establishes that the defect subcategory is Serre and that its perpendicular corresponds to left-exact functors, with connections to Gabriel-Quillen embedding; when $\\\mathcal{C}$ has enough projectives, a precise link to the restricted Yoneda functor emerges and a recollement is obtained. The paper further extends these ideas to direct colimits, showing that Def is localizing and aligns with left-exact functors, thus broadening the Engel-Gabriel framework to extriangulated contexts. Finally, it applies these insights to the general heart construction for cotorsion pairs in triangulated categories, proving the heart is equivalent to the left-exact functors on the cotorsion side and providing a cohomological functor realizing this heart. Overall, the results unify defect theory, left-exact representability, and heart constructions within a common extriangulated setting, with implications for exact/abelian characterizations and embedding theorems.
Abstract
The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{\mathsf{def}}\mathcal{C}\to\operatorname{\mathsf{mod}}\mathcal{C}\to\operatorname{\mathsf{lex}}\mathcal{C}$, where $\operatorname{\mathsf{lex}}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{\mathsf{def}}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel-Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U},\mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is same as the construction of a localization sequence $\operatorname{\mathsf{def}}\mathcal{U}\to\operatorname{\mathsf{mod}}\mathcal{U}\to\operatorname{\mathsf{lex}}\mathcal{U}$.