For $X$ a simplicial set, for $x : \Delta[0] \to X$ a point in $X$, and for $n \in \mathbb{N}$, the $n$th Eilenberg subcomplex$E_n(X,x)$ of $X$ at $x$ is the fiber of the $(n-1)$-coskeleton-projection over $x$, hence the pullback

By the skeleton/coskeletonadjunction$(sk_{n-1} \dashv cosk_{n-1})$ the $n$th Eilenberg subcomplex is the subobject of $X$ consisting of those simplices whose $(n-1)$-skeleton is constant on the point $x$.

Properties

Restriction to Kan complexes

If $X$ is a Kan complex , then so is $E_n(X,x)$ for all $n \in \mathbb{N}$ and $x \in X_0$.

The inclusion $sSet_{(n-1)} \hookrightarrow sSet^{*/}$ of $n$-fold reduced simplicial set (those with a single $k$-simplex for all $k \leq n-1$) into all pointed simplicial sets is a coreflective subcategory with coreflector being forming of the $n$th Eilenberg subcomplex