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**11 - 16**of**16**### HOMOTOPY FIBRE SEQUENCES INDUCED BY 2-FUNCTORS

, 909

"... Abstract. This paper contains some contributions to the study of the relationship between 2-categories 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 2-functors. Mathematical Subject Classif ..."

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Abstract. This paper contains some contributions to the study of the relationship between 2-categories 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 2-functors. Mathematical Subject Classification: 18D05, 55P15, 18F25.

### DIAGONAL FIBRATIONS ARE POINTWISE FIBRATIONS DEDICATED TO THE MEMORY OF SAUNDERS MAC LANE(1909-2005)

, 706

"... On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrat ..."

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On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. 1. Introduction and

### JOURNAL OF PURE AND APPLIED ALGEBRA ELSEVIER Journal of Pure and Applied Algebra 103 (1995) 287-302 Closed model structures for algebraic models of n-types”’

, 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 (2-types), 2-crossed modules of groups (3-types) or, in general. n-hypercrossed complexes of groups ((n + I)-types), as well as that of n-simplicial 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 CW-complexes, with only one

### DIAGONAL FIBRATIONS ARE POINTWISE FIBRATIONS

"... On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrat ..."

Abstract
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On the category of bisimplicial sets there are different Quillen closed model structures associated to various definitions of fibrations. In one of them, which is due to Bousfield and Kan and that consists of seeing a bisimplicial set as a simplicial object in the category of simplicial sets, fibrations are those bisimplicial set maps such that each of the induced simplicial set maps is a Kan fibration, that is, the pointwise fibrations. In another of them, introduced by Moerdijk, a bisimplicial map is a fibration if it induces a Kan fibration of associated diagonal simplicial sets, that is, the diagonal fibrations. In this note, we prove that every diagonal fibration is a pointwise fibration. 1. Introduction and

### Homology, Homotopy and Applications, vol.6(1), 2004, pp.153–166 HOMOLOGY FIBRATIONS AND ”GROUP-COMPLETION” REVISITED

"... We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but fol-lows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and t ..."

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We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but fol-lows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations.