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Journal of Pure and Applied Algebra 180 (2003) 1–23 www.elsevier.com/locate/jpaa
Cooperads and coalgebras as closed model categories Marc Aubrya;∗, David Chataurb aLaboratoire de Mathematiquesà Jean-Alexandre Dieudonneà UMR CNRS No. 6621, Universiteà de Nice, Parc Valrose, 06108 Nice Cedex 02, France bCentre di Recerca Matematica,à Institut d’Estudis Catalans, Apartat 50 E-08193 Bellaterra, Spain
Received 22 October 2001; received in revised form 13 March 2002 Communicated by C. Kassel
Abstract
This paper proves that (linear) quasi-isomorphisms and monomorphisms deÿne a closed model category (CMC) structure on the category of positively graded cooperads. Z-coalgebras over a quasi-cofree cooperad Z supports also an analogous CMC structure. c 2003 Elsevier Science B.V. All rights reserved.
MSC: 18Dxx; 55U35
1. Introduction
The purpose of this paper is to study the homotopy theory of cooperads and of coalgebras over a cooperad. These comultiplicative structures appear naturally in the context of Koszul duality for operads [4]. Thanks to the bar construction for operads they play a key role in the theory of deformation of an algebra over an operad ([7,12], and also [3]). Let us begin with very vague and general re>ections (in these informal lines we use a covariant language and speak of algebras, monads, free objects, etc ::: but the same holds for coalgebras, comonads, etc :::). Whatis meantunder algebraic