E_∞-coalgebras and p-adic homotopy theory

We show that for any separably closed field k of characteristic $$p>0$$ p > 0 , the canonical functor from nilpotent p -adic spaces to E_∞ E ∞ -coalgebras over k (given by singular chains with coefficients in k ) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.