nLab
equivalence in an (infinity,1)-category

Definition

For $C$ a quasi-category, a morphism $f:x\to y$ in $C$ (an edge in the underlying simplicial set) is an equivalence if its image in the homotopy category $\mathrm{Ho}(C)$ is an isomorphism.

Equivalently, $f$ is an equivalence if it is the image of a functor of quasi-categories (i.e. a map of simplicial sets) out of the nerve $N(J)$, where $J$ is the interval groupoid. This is a quasi-categorical version of the general theorem-schema in higher category theory that any equivalence can be improved to an adjoint equivalence.

Related concepts

• equality

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• equivalence in an (∞,1)-category

• equivalence of (∞,1)-categories