strict omega-groupoid

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/ω-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

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.

A **strict $\omega$-groupoid** or **strict $\infty$-groupoid** is a strict ω-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.

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.

There is a model structure on strict ∞-groupoids.

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

A textbook reference is

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.

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