nLab
strict omega-groupoid

Idea

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

By a result by Brown and Higgins, going back to insights by Whitehead, $\omega$-groupoids are equivalently modeled by crossed complexes.

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

$(\mathrm{\infty },1)$-Category of strict $\mathrm{\infty }$-groupoids

There is a model structure on strict ∞-groupoids.

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

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 $\mathrm{\infty }$-groupoids and crossed complexes , Cah. Top. Géom. Diff. 22, 371-386, 1981. (pdf)

Notice that this article says ”$\mathrm{\infty }$-groupoid” for strict globular $\mathrm{\infty }$-groupoid and ”$\omega$-groupoid” for strict cubical $\mathrm{\infty }$-groupoid .