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A Kan fibration that at the same time is a weak homotopy equivalence is called an acyclic Kan fibration. (Also: a trivial Kan fibration.)
Acyclic Kan fibrations are the acyclic fibrations in the classical model structure on simplicial sets , hence those morphisms which have the right lifting property against monomorphisms (degreewise injections) of simplicial sets.
In particular, this implies that:
Proposition 2.1. Acyclic Kan fibrations are (degreewise) surjective, in that they have the right lifting property against any empty bundle .
Remark 2.2. Prop. 2.1 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 2.3. Acyclic Kan fibrations are precisely the morphisms of simplicial sets that have the right lifting property against all simplex boundary inclusions.
Last revised on April 20, 2023 at 17:10:40. See the history of this page for a list of all contributions to it.