Equivalent definitions of pro-unipotent coalgebras I'm trying to find a reference in the literature for equivalence of the following two definitions of pro-unipotent coalgebras.
Definition Let be $H$ a coagumented coalgebra and let $\Delta \colon H \to H \otimes H$ be a coproduct map. $H$ is pro-unipotent if every non-zero comodule $V$ over $H$ has a non-zero invariant vector, i.e. an element $0\ne v\in V$ such that $\rho(v)=v\otimes 1$, where ${\rho\colon V\to V\otimes H}$ is the coaction map.
Definition’ A coaugmented coalgebra $k\to H$ is pro-unipotent if for any element $h\in H$, there is $N\in {\mathbb N}$ such that the value at $h$ of any length $N$ iteration of $\Delta$ belongs to the kernel of the map ${H^{\otimes (N+1)}\to (H/k)^{\otimes (N+1)}}$.
I would be grateful for any account of this equivalence. However, it would be especially amazing to have a textbook exposition of it (and maybe some other properties of pro-unipotent coalgebras).
 A: 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 .
A proof of the claim that any coalgebra (over a field $k$) satisfying your second definition satisfies the first one can be found in my recent preprint https://arxiv.org/abs/2301.09561 , Lemmas 2.1(a) and 2.2(a).
The claim that any coalgebra satisfying your first definition satisfies the second one is, I think, best provable using the structure theory of coassociative coalgebras over a field.
If the maximal cosemisimple subcoalgebra of $H$ (known classically as the coradical of $H$) is different from $k$, then $H$ contains a cosimple subcoalgebra different from the image of the coaugmentation map. The corresponding simple $H$-comodule $V$ is then a counterexample to your first definition.
If $k$ is the maximal cosemisimple subcoalgebra of $H$, then $H$ satisfies your second definition by Proposition 3.4 from my BLMS survey cited above or by Corollary 9.0.4 from Sweedler's book.
