Title: The homotopy theory of coalgebras over simplicial comonads,
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.