[personal profile] posic
I think that, from many points of view, contramodules are much better understood than comodules nowadays. "A contramodule category" is a locally presentable abelian category with a projective generator; this is a very reasonable definition that works well in a number of contexts.

What "a comodule category" is? Is it just a Grothendieck abelian category? A locally presentable abelian category with an injective cogenerator? A hereditary torsion class (localizing Serre subcategory) in the category of modules over an associative ring? A left perpendicular subcategory in the category of modules over an associative ring (in one or another sense of the word)? How are these classes of abelian categories related to one another?

Take a locally presentable abelian category B, an object M in B, the full subcategory Prod(M) of all direct summands of set-indexed products of copies of M in B. Then there exists a unique abelian category A with enough injective objects such that the full subcategory of injective objects in A is equivalent to the full subcategory Prod(M) in B. What can one say about the category A? Does it belong to any of the above-mentioned classes of abelian categories?

Knowing more about contramodules than I know about comodules is in some sense a good indicator of having mastered the subject. Is it?

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