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Definition A pointed simplicial set is a pair $(X,x)$, where $X$ is a simplicial set and $x$ is a vertex of $X$. If $X$ is a Kan complex, then we refer to the pair $(X,x)$ as a pointed Kan complex. If $(X,x)$ and $(Y,y)$ are pointed Kan complex, then a pointed map from $(X,x)$ to $(Y,y)$ is a morphism of Kan complexes $f: X \rightarrow Y$ satisfying $f(x) = y$. We let $\operatorname{Kan}_{\ast }$ denote the category whose objects are pointed Kan complexes and whose morphisms are pointed maps.