Categories of modules, comodules and contramodules over representations

Abstract

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.

On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras

“…In a categorical sense, it may be said therefore that both comodules and contramodules dualize the notion of module over an algebra. Even though the study of contramodules in the literature is not as developed as that of comodules, the topic has seen a lot of renewed interest in recent years (see, for instance, [2], [3], [4], [6], [17], [18], [19], [20], [25]). Accordingly, our notion of an A ∞ -contramodule over an A ∞ -coalgebra C consists of a graded vector space M ∈ V ect Z along with a collection of structure maps…”