The homotopy theory of coalgebras over simplicial comonads (Q1625460)

The purpose of this paper is to explore some consequences of the main result of the authors' previous work [J. Topol. 10, No. 2, 324--369 (2017; Zbl 1381.55010)], which gives conditions under which a model structure on a category \(\mathcal{M}\) can be transferred along adjunctions with (co)domain \(\mathcal{M}\). These results are applied here to create interesting model structures on certain categories of coalgebras belonging to comonads on \(\mathrm{sSet}_*\), the category of pointed simplicial sets. It should be noted that although the proofs in [loc. cit.] are flawed, the statements there are correct, and the errors have been fixed in [\textit{R. Gamer} et al., ``Lifting accessible model structures'', J. Topology 13, No. 1, 59--76 (2020; \url{doi:10.1112/topo.12123})]. The set up is as follows: three examples of comonads on \(sSet_*\) are considered. These arise from \(1)\) the free-forgetful adjunction with the category \(sAb\) of simplicial abelian groups, \(2)\) the simplicial loop-suspension adjunction on \(sSet_*\), \(3)\) the infinite suspension-zero space adjunction with the category \(Sp^\Sigma\) of symmetric spectra. Each of these adjunctions gives rise to a comonad on \(\mathrm{sSet}_*\), universally denoted \(\mathbb{K}\), and an associated category of coalgebras, denoted \(\mathrm{coAlg}_\mathbb{K}\). The authors show that in each case the adjunction is comonadic, i.e. each comparison functor \(Can_\mathbb{K}:\mathrm{coAlg}_\mathbb{K}\rightarrow \mathrm{sSet}_*\) is an equivalence of categories. This equivalence gives rise to a model structure on \(\mathrm{coAlg}_\mathbb{K}\), but the authors show that there is another, more interesting, model structure on this category which is left-induced by the forgetful functor to the other side of the adjunction. With this model structure, the free-forgetful adjunction \(\mathrm{Can}_\mathbb{K}:sSet_*\rightleftarrows \mathrm{coAlg}_\mathbb{K}:V_\mathbb{K}\) is Quillen. More general results are given relating to comonads arising from \(\mathrm{sSet}_*\)-adjunctions between accessible [the authors, loc. cit.] \(\mathrm{sSet}_*\)-model categories. The authors next study the derived counits of the three adjunctions \(\mathrm{Can}_\mathbb{K}:\mathrm{sSet}_*\rightleftarrows \mathrm{coAlg}_\mathbb{K}:V_\mathbb{K}\) and give conditions under which their components are weak equivalences. They show that in case \(1)\), the derived counit of the adjunction is exactly the Bousfield-Kan \(\mathbb{Z}\)-completion, and that in cases \(2)\), \(3)\), the derived counit of the adjunction is a weak equivalence at each fibrant \(\mathbb{K}\)-coalgebra and at each \(1\)-connected simplicial set. The paper includes two appendices so as to make it somewhat self-contained. The first recalls the theory of comonads and their coalgebras, whilst the second summarised the techniques and main results of the authors [loc. cit.] that are need to understand the present paper.