nLab simplicial homotopy equivalence

Idea

A simplicial homotopy equivalence is a homotopy equivalence between simplicial sets.

Definition

A morphism $f\colon A\to B$ of simplicial sets is a simplicial homotopy equivalence if there is a morphism $g\colon B\to A$ and homotopies $p\colon \Delta^1\times A\to A$ from $id_A$ to $g \circ f$ and $q\colon \Delta^1\times B\to B$ from $f \circ g$ to $id_B$ (where $id_X$ denotes the identity morphism on the simplicial set $X$).

Properties

• If $A$ and $B$ are Kan complexes, then $f$ is a simplicial homotopy equivalence if and only if $f$ has the right homotopy lifting property with respect to the boundary inclusion $\partial\Delta^n\to\Delta^n$ of simplices.

• All simplicial homotopy equivalences are simplicial weak equivalences. The converse is true if $A$ and $B$ are Kan complexes.

Related notions

• simplicial weak equivalence

• simplicial Whitehead theorem

• classical model structure on simplicial sets