nLab equivalence of infinity-groupoids

An equivalence of $\infty$-groupoids is equivalently

• an equivalence in the $(\infty,1)$-category ∞Grpd,

• an equivalence of (∞,1)-categories between (∞,1)-categories that happen to be ∞-groupoids,

In terms of presentation by topological spaces or simplicial sets (see at classical model structure on topological spaces and classical model structure on simplicial sets, respectively) this is:

• a weak homotopy equivalence.

If specifically presented by CW-complexes or Kan complexes, respectively, it is

• a homotopy equivalence.