Simplicial presheaves of coalgebras (Q1954170)

Let \(\mathcal{C}\) be a small Grothendieck site and \(\mathrm{PSh}(\mathcal{C})\) the category of set-valued presheaves on \(\mathcal{C}\). Further, let \(\mathcal{R}\) be a presheaf of commutative unital rings on \(\mathcal{C}\), and \(\mathrm{Coalg}_\mathcal{R}\) the category of cocommutative, coassociatve \(\mathcal{R}\)-coalgebras. Finally let \(\mathrm{sCoalg}_\mathcal{R}\), etc., denote the corresponding categories of simplicial objects. This paper studies the relationship between the homotopy theory of \(\mathrm{sCoalg}_\mathcal{R}\) and the category \(\mathrm{sPSh}(\mathcal{C})\) of simplicial presheaves on \(\mathcal{C}\). It is shown that \(\mathrm{sCoalg}_\mathcal{R}\) has a left proper, simplicial, cofibrantly generated model category structure (dependent to some extent on the choice of a large enough regular cardinal). This structure is then related to \(\mathcal{R}\)-local homotopy theory of simplicial presheaves and that of simplicial \(\mathcal{R}\)-modules by various, simply defined, Quillen adjunctions. The special case in which \(\mathcal{R}\) is a presheaf of algebraically closed fields is studied in more detail and, in that case, it is shown that the \(\mathcal{R}\)-local homotopy category of simplicial presheaves embeds in the homotopy category of simplicial \(\mathcal{R}\)-coalgebras. The paper contains a very readable summary of the theory of this area.