**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

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

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:

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.