nLab
strict omega-groupoid

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

A strict ω\omega-groupoid is an algebraic model for certain simple homotopy types/∞-groupoids based on globular sets. It is almost like a chain complex of abelian groups (under Dold-Kan correspondence) except that the fundamental group is allowed, more generally, to be non-abelian and to act on all the other homotopy groups. In fact, strict ω\omega-groupoids are equivalent to crossed complexes.

Definition

A strict ω\omega-groupoid or strict \infty-groupoid is a strict ω-category in which all k-morphisms have a strict inverse for all kk \in \mathbb{N}

Equivalently, it is a globular set X X_\bullet equipped with a unital and associative composition in each degree such that for all pairs of degrees (k 1<k 2)(k_1 \lt k_2) it induces on the 2-graph X k 2X k 1X 0X_{k_2} \stackrel{\to}{\to} X_{k_1} \stackrel{\to}{\to} X_0 the structure of a strict 2-groupoid.

Properties

Relation to crossed complexes

Following work of J. H. C. Whitehead, in (Brown-Higgins) it is shown that the 1-category of strict ω\omega-groupoids is equivalent to that of crossed complexes. This equivalence is a generalization of the Dold-Kan correspondence to which it reduces when restricted to crossed complexes whose fundamental group is abelian and acts trivially. More details in this are at Nonabelian Algebraic Topology.

Strict \infty-groupoids form one of the vertices of the cosmic cube of higher category theory.

Model structure

There is a model structure on strict ∞-groupoids.

This should present the full sub-(∞,1)-category of ∞Grpd of strict \infty-groupoids.

References

A textbook reference is

The equivalence of strict ω\omega-groupoids and crossed complexes is discussed in

Notice that this article says “\infty-groupoid” for strict globular \infty-groupoid and “ω\omega-groupoid” for strict cubical \infty-groupoid, and also contains definitions of nn-fold categories, and of what are now called globular sets.

Revised on July 14, 2013 20:43:03 by Urs Schreiber (82.113.106.95)