Comodules and contramodules over coalgebras associated with locally finite categories

Abstract

We explain how to attach a coalgebra $\mathcal{C}$ over a field $k$ to a small $k$-linear category $\mathsf{E}$ satisfying suitable finiteness conditions. In this context, we study full-and-faithfulness of the contramodule forgetful functor, and describe explicitly the categories of locally finite left $\mathcal{C}$-comodules and left $\mathcal{C}$-contramodules as certain full subcategories of the category of left $\mathsf{E}$-modules.

arXiv commentLaTeX 2e with xy-pic; 35 pages, 1 commutative diagram; v.2: Question 4.12 added, new Section 6 inserted; v.3: misprints corrected, proof of Theorem 3.7 reworked with Lemma 3.8 created, a paragraph inserted in the introduction, Examples 7.6 added at the end