A recollement of differential graded categories
M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez
TL;DR
The paper addresses recollements in differential graded categories by establishing a canonical recollement for a dg-category \(\mathcal{C}\) relative to a full subcategory \(\mathcal{B}\) closed under coproducts, via a dg-ideal \(\mathcal{I}_{\mathcal{B}}\). It develops a robust toolkit based on ends, coends, weighted limits, and Kan extensions (Lan and Ran) to realize adjoints and universal properties in the dg context, and shows how recollements of dg-functor categories induce recollements for triangular matrix dg-categories. A key contribution is proving the existence of the recollement at the dg level and, via underlying abelian categories, obtaining a parallel recollement in the abelian setting. The results generalize prior work by Chen and Zheng to the dg-triangular matrix framework, providing a systematic method to transfer recollement data through enriched categorical constructions with potential applications in representation theory and derived categories.
Abstract
In this paper, we prove that given a differential graded category C and B a full differential graded subcategory closed under coproducts, there is a canonical recollement of differential graded categories, for which we use enriched categories tools. We continue the study of differential graded triangular matrix categories as initiated in [22]. We show that given a recollement between functor dg-categories we can induce a new recollement between differential graded triangular matrix categories, this is a generalization of a result given by Chen and Zheng in [5, Theorem 4.4].