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Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
Semistrict models of connected 3types and Tamsamani’s weak 3groupoids
, 2006
"... Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types usin ..."
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Homotopy 3types can be modelled algebraically by Tamsamani’s weak 3groupoids as well as, in the path connected case, by cat 2groups. This paper gives a comparison between the two models in the pathconnected case. This leads to two different semistrict algebraic models of connected 3types using Tamsamani’s model. Both are then related to Gray groupoids.
SEMISTRICT TAMSAMANI NGROUPOIDS AND CONNECTED NTYPES
, 2007
"... Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of co ..."
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Cited by 5 (1 self)
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Tamsamani’s weak ngroupoids are known to model ntypes. In this paper we show that every Tamsamani weak ngroupoid representing a connected ntype is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tamsamani’s weak ngroupoids and cat n−1groups as models of connected ntypes.
Joins for (Augmented) Simplicial Sets
, 1998
"... We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geomet ..."
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We introduce a notion of join for (augmented) simnplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category \Delta. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geometric) simplicial complexes as given by Brown, [2], or Spanier, [13], reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, [8]) whilst also being applied to the basic properties of polyhedra relating to homology. The theories of geometric and abstract simplicial complexes run in a largely parallel way and when describing the theory, expositions often choose which aspect  abstract combinatorial or geometric  to emphasise at each step. Historically in algebraic topology geometric simplicial complexes, as tools, were largely replaced by CW complexes whilst the combinatorial abstract complex became pa...
Internal categorical structure in homotopical algebra
 Proceedings of the IMA workshop ?nCategories: Foundations and Applications?, June 2004, (to appear). CROSSED MODULES AND PEIFFER CONDITION 135 [Ped95] [Por87
, 1995
"... Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1. ..."
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Abstract. This is a survey on the use of some internal higher categorical structures in algebraic topology and homotopy theory. After providing a general view of the area and its applications, we concentrate on the algebraic modelling of connected (n + 1)types through cat ngroups. 1.
A Homotopy 2Groupoid From a Fibration
, 1999
"... In this paper we give an elementary derivation of a 2groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2groupoids and cat 1 groupoids. ..."
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In this paper we give an elementary derivation of a 2groupoid from a fibration. This extends a previous result for pointed fibrations due to Loday. Discussion is included as to the translation between 2groupoids and cat 1 groupoids.
School of Mathematics,
, 2008
"... We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial ..."
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We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial
JOURNAL OF PURE AND APPLIED ALGEBRA ELSEVIER Journal of Pure and Applied Algebra 103 (1995) 287302 Closed model structures for algebraic models of ntypes”’
, 1993
"... In this paper we give a general method to obtain a closed model structure, in the sense of Quillen, on a category related to the category of simplicial groups by a suitable adjoint situation. Applying this method, categories of algebraic models of connected types such as those of crossed modules of ..."
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In this paper we give a general method to obtain a closed model structure, in the sense of Quillen, on a category related to the category of simplicial groups by a suitable adjoint situation. Applying this method, categories of algebraic models of connected types such as those of crossed modules of groups (2types), 2crossed modules of groups (3types) or, in general. nhypercrossed complexes of groups ((n + I)types), as well as that of nsimplicial groups (all types), inherit such a closed model structure. The problem of giving algebraic models for the homotopy theory of spaces has been studied in the last years by several authors [3,5, 15, 19,201. Classical references about it are the results by Eilenberg and Mac Lane [9] giving the well known equivalence between the homotopy category of pointed connected CWcomplexes, with only one