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A Simplicial Description Of The Homotopy Category Of Simplicial Groupoids
, 2000
"... . In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then chara ..."
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. In this paper we use Quillen's model structure given by DwyerKan for the category of simplicial groupoids (with discrete object of objects) to describe in this category, in the simplicial language, the fundamental homotopy theoretical constructions of path and cylinder objects. We then characterize the associated left and right homotopy relations in terms of simplicial identities and give a simplicial description of the homotopy category of simplicial groupoids. Finally, we show loop and suspension functors in the pointed case. 1. Introduction 1.1. Summary. A wellknown and quite powerful context in which an abstract homotopy theory can be developed is supplied by a category with a closed model structure in the sense of Quillen [16]. The category Simp(Gp) of simplicial groups is a remarkable example of what a closed model category is, and the homotopy theory in Simp(Gp) developed by Kan [12] occurs as the homotopy theory associated to this closed model structure. According to the t...
Closed Model Categories For [n,m]Types
, 1997
"... For m ? n ? 0, a map f between pointed spaces is said to be a weak [n; m]equivalence if f induces isomorphisms of the homotopy groups ß k for n 6 k 6 m . Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the sa ..."
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For m ? n ? 0, a map f between pointed spaces is said to be a weak [n; m]equivalence if f induces isomorphisms of the homotopy groups ß k for n 6 k 6 m . Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but different classes of fibrations and therefore of cofibrations. Using one of these structures, one obtains that the localized category is equivalent to the category of nreduced CW  complexes with dimension less than or equal to m+ 1 and mhomotopy classes of cellular pointed maps. Using the other structure we see that the localized category is also equivalent to the homotopy category of (n  1)connected (m + 1)coconnected CW complexes.
THE ALGEBRAIC KTHEORY OF A DIAGRAM OF RINGS
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL. 10(2), 2008, PP.13–58
, 2008
"... In this paper, we consider “diagrams of rings”, or functors ..."
CLOSED MODEL CATEGORIES FOR [n, m]TYPES
"... Abstract. For m � n> 0, a map f between pointed spaces is said to be a weak [n, m]equivalence if f induces isomorphisms of the homotopy groups πk for n � k � m. Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures h ..."
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Abstract. For m � n> 0, a map f between pointed spaces is said to be a weak [n, m]equivalence if f induces isomorphisms of the homotopy groups πk for n � k � m. Associated with this notion we give two different closed model category structures to the category of pointed spaces. Both structures have the same class of weak equivalences but different classes of fibrations and therefore of cofibrations. Using one of these structures,