nLab Eilenberg subcomplex

Definition

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

$$\array{ E_n(X,x) &\to& X \\ \downarrow && \downarrow \\ * &\stackrel{x}{\to}& cosk_{n-1}X } \,.$$

By the skeleton/coskeleton adjunction $(sk_{n-1} \dashv cosk_{n-1})$ 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 $x \in X_0$.

Relation to $n$-connected objects

If $X$ is a Kan complex and (n-1)-connected, then the canonical morphism $E_n(X,x) \to X$ is a homotopy equivalence.

See (May, theorem 8.4).

Relation to pointed $n$-connected objects

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

$$sSet^{*/} \underoverset {\underset{E_n(-,*)}{\longrightarrow}} {\overset{}{\hookleftarrow}} {\bot} sSet_{n-1} \,.$$

the counit of this adjunction is the defining inclusion $E_n(X,*) \to X$.

So if $(* \to X) \in sSet^{*/}$ such that $X \in sSet$ is a Kan complex and (n-1)-connected, then the counit $E_n(X,*) \to X$ is a homotopy equivalence.

Accordingly, the coreflection presents the inclusion of (n-1)-connected pointed infinity-groupoids into all pointed infinity-groupoids

$$\infty Grpd_{\geq (n-1)}^{*/} \hookrightarrow \infty Grpd^{*/} \,.$$

Related concepts

• Whitehead tower, Postnikov tower

References

Around def. 8.3 in

• Peter May, Simplicial objects in algebraic topology (djvu)