Double Groupoids and Homotopy 2-types

作者：Martinez Cegarra Antonio; Heredia Benjamin A; Remedios Josue

来源：Applied Categorical Structures, 2012, 20(4): 323-378.

DOI：10.1007/s10485-010-9240-1

• Double groupoid
• Classifying space
• Bisimplicial set
• Kan complex
• Geometric realization
• Homotopy type

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, C R 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 %26apos;filling condition%26apos;. Any such double groupoid characteristically has associated to it %26apos;homotopy groups%26apos;, which are defined using only its algebraic structure. Thus arises the notion of %26apos;weak equivalence%26apos; between such double groupoids, and a corresponding %26apos;homotopy category%26apos; 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 homotopy group at any base point vanishes for n a parts per thousand yenaEuro parts per thousand 3). A quasi-inverse functor is explicitly given by means of a new %26apos;homotopy double groupoid%26apos; construction for topological spaces.

• 出版日期2012-8