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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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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).
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
A full and faithful nerve for 2categories
 Appl. Categ. Structures
, 2005
"... We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We al ..."
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We prove that there is a full and faithful nerve functor defined on the category 2Catlax of 2categories and (normal) lax 2functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the socalled geometric nerve of a 2category or of a 2groupoid. We also show that (normal) lax 2natural transformations produce homotopies of a special kind, and that two lax 2functors from a 2category to a 2groupoid have homotopic nerves if and only if there is a lax 2natural transformation between them. 1
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.