The homotopy theory of coalgebras over simplicial comonads

TL;DR

This paper establishes model category structures for coalgebras over simplicial comonads, providing explicit fibrant replacements and analyzing the derived unit and counit maps in specific examples, advancing the homotopy theory of coalgebras.

Contribution

It applies the Acyclicity Theorem to construct model structures on coalgebras over simplicial comonads and analyzes the derived unit and counit maps in concrete cases.

Findings

Derived counit components are isomorphisms at fibrant objects.

Derived unit components are weak equivalences at 1-connected objects.

In one example, the derived unit equals the Bousfield-Kan completion map.

Abstract

We apply the Acyclicity Theorem of Hess, Kerdziorek, Riehl, and Shipley (recently corrected by Garner, Kedziorek, and Riehl) to establishing the existence of model category structure on categories of coalgebras over comonads arising from simplicial adjunctions, under mild conditions on the adjunction and the associated comonad. We study three concrete examples of such adjunctions where the left adjoint is comonadic and show that in each case the component of the derived counit of the comparison adjunction at any fibrant object is an isomorphism, while the component of the derived unit at any 1-connected object is a weak equivalence. To prove this last result, we explain how to construct explicit fibrant replacements for 1-connected coalgebras in the image of the canonical comparison functor from the Postnikov decompositions of their underlying simplicial sets. We also show in one case that the derived unit is precisely the Bousfield-Kan completion map.