  A classical result of Tannaka duality is the fact that a coalgebra over a
field can be reconstructed from its category of finite dimensional
representations by using the forgetful functor which sends a representation to
its underlying vector space. There is also a corresponding recognition result,
which characterizes those categories equipped with a functor to finite
dimensional vector spaces which are equivalent to the category of finite
dimensional representations of a coalgebra.
  In this paper we study a generalization of these questions to an arbitrary
cosmos, that is, a complete and cocomplete symmetric monoidal closed category.
Instead of representations on finite dimensional vector spaces we look at
representations on objects of the cosmos which have a dual. We give a necessary
and sufficient condition that ensures that a comonoid can be reconstructed from
its representations, and we characterize categories of representations of
certain comonoids. We apply this result to certain categories of filtered
modules which are used to study p-adic Galois representations.
