Point-set models for homotopy coherent coalgebras

Abstract

We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on the other, we give a general definition of an $\infty$-category of coalgebras over an enriched $\infty$-operad. We show by induction over cell attachments that these two $\infty$-categories are in fact equivalent when the operad is cofibrant. This yields explicit point-set models for $\mathbb{E}_n$-coalgebras and $E_\infty$-coalgebras in the derived $\infty$-category of chain complexes over a field, and an explicit point-set model for the cellular chains functor with its $E_\infty$-coalgebra structure. After Bachmann--Burklund, this gives a point-set algebraic model for nilpotent $p$-adic homotopy types.