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.

Remark

In particular, this implies that acyclic Kan fibrations are always (in particular: degreewise) surjective in that they have the right lifting property against any empty bundle $\varnothing \xhookrightarrow{\;} S$ (this is in contrast to plain Kan fibrations, see this remark).

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