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What you call "pro-unipotent coalgebras" are called "pointed irreducible coalgebras" in classical M.E. Sweedler's book "Hopf algebras" (W.A. Benjamin, New York, 1969), Section 8.0. … I call them "conilpotent coalgebras"; see e.g. Section 3 of the survey paper https://doi.org/10.1112/blms.12797 . …

Consider the Lie algebra of vector fields on the formal disk, $\mathfrak g=k[[t]]d/dt$, where $k$ is a field of characteristic zero and $k[[t]]$ is the $k$-algebra of formal Taylor power series in the …

Any morphism of coaugmented coalgebras $\phi\colon X\to Y$ preserves the canonical filtrations, i.e., one has $\phi(F_nX)\subset F_nY$ for any morphism of coaugmented coalgebras $\phi$. … Accordingly, you can define a morphism of conilpotent coalgebras $\phi\colon X\to Y$ as an arbitrary morphism of coalgebras $X\to Y$, or if you prefer, as a morphism of coaugmented coalgebras $X\to Y$. …

So one cannot perform this construction for coalgebras, generally speaking, but only for topological rings. … Still there are plenty of coalgebras $C$ for which the forgetful functor $C{-}\mathbf{contra}\to C^*{-}\mathbf{mod}$ is already fully faithful. …

But it does hold for surprisingly many infinite-dimensional coalgebras $C$ or topological rings $R$. (In particular, it certainly holds for $C^*=k[[t]]$.) …

This answer concerns contramodules that are finitely generated as objects of the category of contramodules, in the sense of the abstract category-theoretic definition of "finitely generated". The exa …

For any (coassociative, counital) coalgebra $C$ over a field $k$, there is a fully faithful exact functor from the category of $C$-comodules to the category of $C^*$-modules. The essential image of t …

For coassociative dg-coalgebras over any field $k$ the answer is positive, because: Let $C$ be a $\mathbb Z$-graded coalgebra and $D\subset C$ a finite-dimensional ungraded subcoalgebra (of the underlying … Indeed, even for ungraded coalgebras over a field of characteristic $0$, there is an example of infinite-dimensional Lie coalgebra $L$ having no nonzero finite-dimensional subcoalgebras. …

The tensor product of two conilpotent coassociative dg-coalgebras over any field $k$ is a conilpotent coassociative dg-coalgebra over $k$. … Then the question reduces to showing that the tensor product of two conilpotent coalgebras is a conilpotent coalgebra. …

I am not sure that I really understand what you want, but I'd say the relevant structure is that of a module category over a monoidal category. Given a monoidal category $\mathcal E$, one can conside …

Define a cosimple coalgebra as a coalgebra having no nonzero proper subcoalgebras, and a cosemisimple coalgebra as a direct sum of cosimple coalgebras. …

Let us replace "dualizable" with "Noetherian". Then, I believe, for any locally Noetherian Grothendieck (abelian) category with an exact, associative tensor product functor preserving direct limits ( …

So the category of coalgebras over a field k is the category of ind-objects in the category of finite-dimensional coalgebras, while the latter is the opposite category to the category of finite-dimensional … sum of the coalgebras sitting at the points of this set. …