#### Volume 13, issue 4 (2013)

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Simplicial presheaves of coalgebras

### George Raptis

Algebraic & Geometric Topology 13 (2013) 1967–2000
##### Abstract

The category of simplicial $\mathsc{ℛ}$–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 $\mathsc{ℛ}$–local homotopy theory of simplicial presheaves and the homotopy theory of simplicial $\mathsc{ℛ}$–modules by Quillen adjunctions. We study the comparison with the $\mathsc{ℛ}$–local homotopy theory of simplicial presheaves in the special case where $\mathsc{ℛ}$ is a presheaf of algebraically closed (or perfect) fields. If $\mathsc{ℛ}$ is a presheaf of algebraically closed fields, we show that the $\mathsc{ℛ}$–local homotopy category of simplicial presheaves embeds fully faithfully in the homotopy category of simplicial $\mathsc{ℛ}$–coalgebras.

##### Keywords
coalgebras, simplicial presheaves, local homotopy theory, combinatorial model category
##### Mathematical Subject Classification 2010
Primary: 18F20, 18G30, 55U35, 16T15
##### Publication
Received: 24 September 2012
Accepted: 15 January 2013
Published: 29 May 2013
##### Authors
 George Raptis Universität Osnabrück Institut für Mathematik Albrechtstrasse 28a D-49076 Osnabrück Germany