nLab
infinity-groupoid

An $\mathrm{\infty }$-groupoid – equivalently an (∞,0)-category is an ∞-category in which all morphisms are equivalences.

Models

Familiar models for the category ∞-Grpd of weak $\mathrm{\infty }$-groupoids are

• simplicial sets equipped with the standard model structure on simplicial sets;

• Kan complexes

• CW-complexes; see homotopy theory.

Strict $\mathrm{\infty }$-groupoids

Entirely strict $\mathrm{\infty }$-groupoids, usually called $\omega$-groupoids or strict ω-groupoids, are equivalent to crossed complexes.