Simplicial presheaves of coalgebras

TL;DR

This paper develops a model category structure for simplicial R-coalgebras over a presheaf of rings, linking it to local homotopy theories and embedding certain categories fully faithfully.

Contribution

It introduces a new model structure on simplicial R-coalgebras and explores its relationship with local homotopy theories of simplicial presheaves.

Findings

The model category is left proper, simplicial, cofibrantly generated.

The model structure relates to R-local homotopy theory via Quillen adjunctions.

Full embedding of R-local homotopy category into simplicial R-coalgebras for algebraically closed fields.

Abstract

The category of simplicial R-coalgebras over a presheaf of commutative unital rings on a small Grothendieck site is endowed with a left proper, simplicial, cofibrantly generated model category structure where the weak equivalences are the local weak equivalences of the underlying simplicial presheaves. This model category is naturally linked to the R-local homotopy theory of simplicial presheaves and the homotopy theory of simplicial R-modules by Quillen adjunctions. We study the comparison with the R-local homotopy category of simplicial presheaves in the special case where R is a presheaf of algebraically closed (or perfect) fields. If R is a presheaf of algebraically closed fields, we show that the R-local homotopy category of simplicial presheaves embeds fully faithfully in the homotopy category of simplicial R-coalgebras.