nLab acyclic Kan fibration

Definition

A Kan fibration that at the same time is a weak homotopy equivalence is called an acyclic Kan fibration. (Also: a trivial Kan fibration.)

Properties

Acyclic Kan fibrations are the acyclic fibrations in the classical model structure on simplicial sets $sSet_{Qu}$, hence those morphisms which have the right lifting property against monomorphisms (degreewise injections) of simplicial sets.

In particular, this implies that:

Proposition

Acyclic Kan fibrations are (degreewise) surjective, in that they have the right lifting property against any empty bundle $\varnothing \xhookrightarrow{\;} S$.

Remark

Prop. is in contrast to plain Kan fibrations, see this remark about the empty horn, and see the discussion on surjective Kan fibrations there.

In fact:

Proposition

Acyclic Kan fibrations are precisely the morphisms of simplicial sets that have the right lifting property against all simplex boundary inclusions.

See this Prop. at classical model structure on simplicial sets. This is part of the statement that $sSet_{Qu}$ is a cofibrantly generated (see this Prop.).

Related entries

• simplicial homotopy theory