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

$\xymatrix{ \Lambda _ k[n] \ar[r] \ar[d] & X \ar[d] \\ \Delta [n] \ar[r] \ar@{-->}[ru] & Y }$

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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