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The limit in the category of graded abelian hopf algebras over $ \mathbb{F}_p$ forgets to the limit in the category of graded cocommutative coalgebras over $\mathbb{F}_p$ but does generally not forget … limit in graded cocommutative coalgebras over $\mathbb{F}_p$. …

Question: Can one obtain $C$ by iterately taking pushouts of trivial coalgebras (= coalgebras with zero comultiplication)? … Especially, does $C$ belong to the full subcategory generated by trivial coalgebras under small colimits? If not, can anyone give a counterexample? …

But taking the cokernel of the coaugmentation only defines an oplax symmetric monoidal functor from cocommutative coaugmented counital dg-coalgebras to cocommutative non-counital dg-coalgebras. … as monoids in the category of conilpotent cocommutative coaugmented counital dg-coalgebras. …

\to C_*(K(G_2,n);F) \to C_*(K(G_1,n);F) \ (*)$$ on chains with F-coeffients seen as inverse system of conilpotent coassociative dg-coalgebras over $F$ has inverse limit $C_*(K(G,n);F).$ Is there a left … induced model structure on the category of conilpotent coassociative dg-coalgebras over $F$ such that $C_*(K(G,n);F)$ is the homotopy inverse limit of the diagram $(*)$? …

We think of the $\mathrm{E}(\mathrm{X})$ as the most basic conilpotent cocommutative non-counital dg-coalgebras. … implicitely use that the model structure on chain complexes over $\mathrm{K}$ with weak equivalences the quasi-isos lifts to a model structure on the category of conilpotent cocommutative non-counital dg-coalgebras …