Article

# Double Groupoids and Homotopy 2-types

Applied Categorical Structures (Impact Factor: 0.38). 04/2012; DOI:10.1007/s10485-010-9240-1
Source: arXiv

ABSTRACT This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types
of their classifying spaces. Double categories (Ehresmann, CR Acad Sci Paris 256:1198–1201, 1963a, Ann Sci Ec Norm Super 80:349–425, 1963b) have well-understood geometric realizations, and here we deal with homotopy types represented by double groupoids satisfying
a natural ‘filling condition’. Any such double groupoid characteristically has associated to it ‘homotopy groups’, which are
defined using only its algebraic structure. Thus arises the notion of ‘weak equivalence’ between such double groupoids, and
a corresponding ‘homotopy category’ is defined. Our main result in the paper states that the geometric realization functor
induces an equivalence between the homotopy category of double groupoids with filling condition and the category of homotopy
2-types (that is, the homotopy category of all topological spaces with the property that the

n\textthn{\text{th}} homotopy group at any base point vanishes for n ≥ 3). A quasi-inverse functor is explicitly given by means of a new ‘homotopy double groupoid’ construction for topological
spaces.

KeywordsDouble groupoid-Classifying space-Bisimplicial set-Kan complex-Geometric realization-Homotopy type
Mathematics Subject Classifications (2010)18D05-20L05-55Q05-55U40

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