$$\mathbb {E}_\infty $$-coalgebras and p-adic homotopy theory

Abstract We show that for any separably closed field k of characteristic $$p&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , the canonical functor from nilpotent p -adic spaces to $$\mathbb {E}_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -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.