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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.
The Classifying Space of a Topological 2Group
, 2008
"... Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal G ..."
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Cited by 4 (1 self)
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Categorifying the concept of topological group, one obtains the notion of a ‘topological 2group’. This in turn allows a theory of ‘principal 2bundles’ generalizing the usual theory of principal bundles. It is wellknown that under mild conditions on a topological group G and a space M, principal Gbundles over M are classified by either the Čech cohomology ˇ H 1 (M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2groups and even topological 2categories. We explain various viewpoints on topological 2groups and the Čech cohomology ˇ H 1 (M, G) with coefficients in a topological 2group G, also known as ‘nonabelian cohomology’. Then we give an elementary proof that under mild conditions on M and G there is a bijection ˇH 1 (M, G) ∼ = [M, BG] where BG  is the classifying space of the geometric realization of the nerve of G. Applying this result to the ‘string 2group ’ String(G) of a simplyconnected compact simple Lie group G, it follows that principal String(G)2bundles have rational characteristic classes coming from elements of H ∗ (BG, Q)/〈c〉, where c is any generator of H 4 (BG, Q).
Localizations of Transfors
, 1998
"... Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, doe ..."
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Let C , D and E be ndimensional teisi, i.e., higherdimensional Graycategorical structures. The following questions can be asked. Does a left qtransfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right qtransfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, does a functor C\Omega D ! E induce a functor D\Omega C ! E? For c; c 0 elements of C whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C ! D induce a qtransfor C (c;c 0 ) ! D (d;d 0 ), for appropriate d;d 0 2 D ? For c; c 0 2 C and d;d 0 2 D whose (k \Gamma 1)sources and (k \Gamma 1)targets agree, does a qtransfor C\Omega D ! E induce a (q+k+1)transfor C (c;c 0 )\Omega D (d;d 0 ) ! E(e;e 0 ), for appropriate e; e 0 2 E? I give answers to these questions in the cases where ndimensional teisi and their tensor product have been defined, i.e., for n 3, and in some cases for n up to 5 which do not need all data and axioms...
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.
NERVES AND CLASSIFYING SPACES FOR BICATEGORIES
, 2009
"... This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate ‘nerves of C ’ are homotopy equivalent. Any one of these realiza ..."
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This paper explores the relationship amongst the various simplicial and pseudosimplicial objects characteristically associated to any bicategory C. It proves the fact that the geometric realizations of all of these possible candidate ‘nerves of C ’ are homotopy equivalent. Any one of these realizations could therefore be taken as the classifying space BC of the bicategory. Its other major result proves a direct extension of Thomason’s ‘Homotopy Colimit Theorem’ to bicategories: When the homotopy colimit construction is carried out on a diagram of spaces obtained by applying the classifying space functor to a diagram of bicategories, the resulting space has the homotopy type of a certain bicategory, called the ‘Grothendieck construction on the diagram’. Our results provide coherence for all reasonable extensions to bicategories of Quillen’s definition of the ‘classifying space ’ of a category as the geometric realization of the category’s Grothendieck nerve, and they are applied to monoidal (tensor) categories through the elemental ‘delooping’ construction.
HOMOTOPY FIBRE SEQUENCES INDUCED BY 2FUNCTORS
, 909
"... Abstract. This paper contains some contributions to the study of the relationship between 2categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2functors. Mathematical Subject Classif ..."
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Abstract. This paper contains some contributions to the study of the relationship between 2categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2functors. Mathematical Subject Classification: 18D05, 55P15, 18F25.
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"... Closed model categories for presheaves of simplicial groupoids and presheaves of 2groupoids by Zhiming Luo ..."
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Closed model categories for presheaves of simplicial groupoids and presheaves of 2groupoids by Zhiming Luo
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"... Closed model categories for presheaves of simplicial groupoids and presheaves of 2groupoids by ZhiMing Luo ..."
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Closed model categories for presheaves of simplicial groupoids and presheaves of 2groupoids by ZhiMing Luo
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