I would like to know if an answer to the following question has already been worked out and is perhaps already somewhere in the category theory literature: suppose C is an abelian category and U is a comonad on C. I would like to know if there is a characterization of which objects of C admit the structure of a U-coalgebra. One can easily get partway to an answer to this question: for an object X of C to be a U-coalgebra, we need a counital, coassociative morphism from X to UX. If one only wants to know whether X admits a counital morphism from X to UX, one can pretty easily write down a class in an Ext^1 group which is zero if and only if X admits such a counital morphism (Nuss calls this class an "Atiyah obstruction" in his paper on "Noncommutative descent"). So really what I want to know is the answer to the rest of the question: suppose that X is an object of C which admits a counital morphism from X to UX. Is there any known simple test or characterization for whether X admits a counital _and also coassociative_ morphism from X to UX? Thanks, Andrew S. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]