Quantum topology studies nasty filtered objects. For instance, one can construct a filtration on the infinite-dimensional vector space spanned by the set of tangles on n-points by counting the number of strand crossing changes required to zero-out vectors. There are certain functors which, from nasty filtered objects, produce nice filtered objects. To solve the "formality problem" associated with a nasty object is to find a structure preserving map from the nasty object into the corresponding nice object.
In this talk we discuss solving formality problems as finding (Eilenberg-Moore-esk) coalgebras for idempotent semi-comonads, and use (co)kernels in the category of coalgebras to solve a certain formality problem.