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Let C be a small category. In this paper, we mainly study the category of modules Mod-R on ringed sites (C, R). We firstly reprove the Theorem A of the paper (M. Wu and F. Xu. Skew category algebras and modules on ringed finite sites. J. A. 631, 2023), then we characterize Mod-R in terms of the torsion presheaves on Gr (R), where Gr (R) is the linear Grothendieck construction of R. Finally, we investigate the hereditary torsion pairs, TTF triples and Abelian recollements of Mod-R. When C is finite, the complete classifications of all these are given respectively.
Mawei Wu (Fri,) studied this question.