Equivalence of categories, Gruson-Jensen duality, and applications

For coalgebras $C$ over a field, we study when the categories ${}^C\Mm$ of left $C$-comodules and $\Mm^C$ of right $C$-comodules are symmetric categories, in the sense that there is a duality between the categories of finitely presented unitary left $R$-modules and finitely presented unitary left $L$-modules, where $R$ and $L$ are the functor rings associated to the finitely accessible categories ${}^C\Mm$ and $\Mm^C$.

💡 Research Summary

The paper investigates when the left‑comodule category ({}^C!\mathsf{M}) and the right‑comodule category (\mathsf{M}^C) of a coalgebra (C) over a field are “symmetric” in the sense of Gruson‑Jensen duality. For any finitely accessible category (\mathcal{A}) one can form its functor ring (R=\operatorname{End}_{\mathcal{A}}(U)^{\mathrm{op}}) where (U) is a coproduct of a set of representatives of the finitely presented objects. The finitely presented left (R)‑modules (\mathrm{fp}({}_R!\mathsf{Mod})) are then equivalent to the finitely presented objects of (\mathcal{A}). The same construction applied to (\mathcal{B}=\mathsf{M}^C) yields a ring (L). The central question is whether there is a duality \