Weakly globular \(\text{cat}^n\)-groups and Tamsamani's model
type: Item
instance of: scholarly article
title: Weakly globular \(\text{cat}^n\)-groups and Tamsamani's model
zbMATH Open document ID: 1173.55005
author: Simona Paoli
published in: Advances in Mathematics
publication date: 2009-08-21T00:00:00Z
review text: This long and well-written paper makes several important contributions to the study of algebraic models for connected homotopy \((n+1)\)-types, i.e., connected CW complexes with vanishing homotopy groups in degrees higher than \(n+1.\) The author introduces a subcategory \(\text{Cat}^n(\text{GP})_S\) of Loday's category of \(\text{cat}^n\)-groups [\textit{J.-L. Loday}, J. Pure Appl. Algebra 24, 179--202 (1982; Zbl 0491.55004)]. She shows that these ``special'' \(\text{cat}^n\)-groups model connected \((n+1)\)-types and provide natural algebraic formulations for weak-equivalences and Postnikov decompositions. Special \(\text{cat}^n\)-groups are then compared to Tamsamani's model of weak \((n+1)\)-groupoids [\textit{Z. Tamsamani}, K-Theory 16, No.~1, 51--99 (1999; Zbl 0934.18008)]. The author constructs a functor from the category of \(\text{cat}^n\)-groups to Tamsamani's category \(\mathcal{T}_{n+1}\) and shows this functor factors through \(\text{Cat}^n(\text{GP})_S\). This comparison functor preserves homotopy type. A consequence of this fact is a ``semistrictification'' result for \(\mathcal{T}_{n+1}\). The author identifies a full subcategory \(\mathcal{S}_{n+1}\) of \(\mathcal{T}_{n+1}\) and proves that every \(\mathcal{G} \in \mathcal{T}_{n+1}\) with \(B\mathcal{G}\) connected is linked by a zig-zag of \((n+1)\)-equivalences to an object of \(\mathcal{S}_{n+1}\). The category \(\mathcal{S}_{n+1}\) is semistrict in the sense it cannot be strictified while preserving homotopy type. The details of these constructions and results are extensive. The author does an excellent job of motivating and providing background making the paper relatively self-contained.
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 18D05
zbMATH DE Number: 5597915
zbMATH Keywords: homotopy types
zbMATH Keywords: weak \(n\)-groupoid
zbMATH Keywords: cat\(^n\)-groups
reviewed by: Samuel Bruce Smith
MaRDI profile type: Publication
full work available at URL: j.aim.2009.05.013
OpenAlex ID: W2000944063
cites work: Higher-dimensional algebra. III: \(n\)-categories and the algebra of opetopes
cites work: Monoidal globular categories as a natural environment for the theory of weak \(n\)-categories
cites work: Introduction to bicategories
cites work: Q4304740
cites work: Groupoids and crossed objects in algebraic topology
cites work: Van Kampen theorems for diagrams of spaces
cites work: On cat\(^ n\)-groups and homotopy types
cites work: (Co)Homology of crossed modules
cites work: Derived functors and the homology of \(n\)-types
cites work: Homology of \((n+1)\)-types and Hopf type formulas
cites work: Comparing operadic theories of \(n\)-category
cites work: Q3594561
cites work: Q3424496
cites work: Higher-dimensional crossed modules and the homotopy groups of \((n+1)\)- ads
cites work: Q4001605
cites work: Q5701014
cites work: Localization Theories for Simplicial Presheaves
cites work: Coherence for tricategories
cites work: The ``closed subgroup theorem'' for localic herds and pregroupoids
cites work: Q5431519
cites work: Weak identity arrows in higher categories
cites work: A survey of definitions of \(n\)-category
cites work: Spaces with finitely many non-trivial homotopy groups
cites work: Q5655527
cites work: On the 3-Type of a Complex
cites work: Q3374890
cites work: Q5550390
cites work: Algebraic classification of equivariant homotopy 2-types. I
cites work: Obstruction theory in algebraic categories. II
cites work: Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids
cites work: Extensions, crossed modules and internal categories in categories of groups with operations
cites work: \(n\)-types of simplicial groups and crossed \(n\)-cubes
cites work: Spectral sequences of a double semi-simplicial group
cites work: Homotopical algebra
cites work: Classifying spaces and spectral sequences
cites work: Nonstrict notions of \(n\)-category and \(n\)-groupoid via multisimplical sets
cites work: Q4296949
DOI: 10.1016/J.AIM.2009.05.013
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description: scientific article; zbMATH DE number 5597915