An equivalence in an ∞-category is a morphism f: A → B that has an inverse up to higher homotopies: there exists g: B → A with g ∘ f homotopic to id_A and f ∘ g homotopic to id_B, together with coherent higher homotopies witnessing these identifications. In an ordinary category, this reduces to isomorphism; in higher categories, the higher coherence data is essential.
An equivalence of categories (ordinary or ∞) is a functor F: C → D that is essentially surjective (every object of D is equivalent to some F(c)) and fully faithful (F induces equivalences on all hom-spaces). Equivalent categories have the same categorical structure — they may differ in the specific objects they contain, but not in the patterns of morphisms between them.
Equivalence is the correct notion of “sameness” in higher category theory. Demanding strict equality (isomorphism) of ∞-categories is both too restrictive and poorly behaved; equivalence allows the flexibility that higher-dimensional structure requires. In homotopy type theory, the univalence axiom asserts that equivalent types are equal, making equivalence the foundational notion of identity.