existence of descent data/coalgebras over a comonad

existence of descent data/coalgebras over a comonad

[Andrew Salch]

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.


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