Wakamatsu-tilting subcategories in extriangulated categories
Zhiwei Zhu, Jiaqun Wei
TL;DR
The paper extends tilting theory to extriangulated categories with enough projectives and injectives by introducing Wakamatsu-tilting and Wakamatsu-cotilting subcategories, and proves these notions are equivalent and align with the $\infty$-tilting/$\infty$-cotilting frameworks. It develops a comprehensive recollement-based approach, showing that Wakamatsu-tilting subcategories can be glued from two outer categories into the middle one, and under certain exactness and compatibility conditions, the converse descent holds. The work unifies WT, WC, and $\infty$-tilting/$\infty$-cotilting concepts, extending their properties to the extriangulated setting and providing practical tools for constructing and deconstructing tilting substructures via recollements. This has potential implications for representation theory and homological algebra where extriangulated categories generalize both exact and triangulated contexts. Key contributions include the equivalence of WT/WC with $\infty$-tilting$/$\infty$-cotilting and the explicit gluing/converse results in recollements.
Abstract
Let $\mathscr{C}$ be an extriangulated category with enough projectives and injectives. We give the definitions of Wakamatsu-tilting subcategories and Wakamatsu-cotilting subcategories of $\mathscr{C}$ and show that they coincide with each other. Moreover, the definitions of $\infty$-tilting subcategories and $\infty$-cotilting subcategories given by Zhang, Wei and Wang also coincide with them. As a result, Wakamatsu-tilting subcategories success all properties of $\infty$-tilting subcategories and $\infty$-cotilting subcategories. On the other hand, we glue the Wakamatsu-tilting subcategories in a special recollement and show that the converse of the gluing holds under certain conditions.