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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 acyclic Kan fibrations are always (in particular: degreewise) surjective in that they have the right lifting property against any empty bundle (this is in contrast to plain Kan fibrations, see this remark).
In fact:
Acyclic Kan fibrations are precisely the morphisms of simplicial sets that have the right lifting property against all simplex boundary inclusions.
Last revised on July 21, 2022 at 15:42:47. See the history of this page for a list of all contributions to it.