Definition 14.31.1. A map $X \to Y$ of simplicial sets is called a *Kan fibration* if for all $k, n$ with $1 \leq n$, $0 \leq k \leq n$ and any commutative solid diagram

a dotted arrow exists making the diagram commute. A *Kan complex* is a simplicial set $X$ such that $X \to *$ is a Kan fibration, where $*$ is the constant simplicial set on a singleton.

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