A characterization of modules over dg-representations of small categories
Mawei Wu
TL;DR
The paper studies modules over dg-representations of small categories by gluing right dg-modules via the linear Grothendieck construction. It proves a dg-version of a known gluing result, giving an equivalence ${\rm Mod-}R \simeq (Gr(R)^{op}, {\rm Ch}_{dg}(k))$ and showing ${\rm Mod-}R$ is a Grothendieck category with a projective generator, with Freyd-type Ab-valued descriptions. Leveraging this characterization, the authors classify hereditary torsion pairs, (split) TTF triples, and Abelian recollements in ${\rm Mod-}R$ through explicit combinatorial data: linear Grothendieck topologies on a generator subcategory, idempotent ideals, and center idempotents. Overall, the work extends Estrada–Virili style results to the dg setting, providing a robust framework for localization and recollement in dg-representations of small categories and enabling systematic structural analysis in this context.
Abstract
Let $\mathcal{C}$ be a small category and let $R$ be a dg-representation of the category $\mathcal{C}$, that is, a pseudofunctor from a small category to the category of small dg $k$-categories, where $k$ is a commutative unital ring. In this paper, we mainly study the category $\mbox{Mod-} R$ of right modules over $R$. We characterize it as an ordinary category of dg-modules over a (differential graded) dg-category $Gr(R)$, where $Gr(R)$ is the linear Grothendieck construction of $R$. This characterization generalizes the Theorem 3.18 of the paper (S. Estrada and S. Virili. Cartesian modules over representations of small categories. Adv. in Math. 310: 557-609, 2017) of Estrada and Virili to the dg-category context. Furthermore, as some applications of the main characterization theorem, we classify the hereditary torsion pairs, (split) TTF triples and Abelian recollements in $\mbox{Mod-} R$ respectively.