nLab strict omega-groupoid

Idea

A strict $\omega$-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 $\omega$-groupoids are equivalent to crossed complexes.

The strict $\omega$-groupoids form an (∞,1)-category StrωGrpd.

Definition

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

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

Related concepts

• weak ∞-groupoid, weak ∞-category

• hypercrossed complex

References

A textbook reference is

• Ronnie Brown, Philip Higgins, Rafael Sivera, Nonabelian Algebraic Topology

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

• Ronnie Brown, Philip Higgins, The equivalence of $\infty$-groupoids and crossed complexes , Cah. Top. Géom. Diff. 22, 371-386, 1981. (pdf)

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