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 subcompolex $E_n(X,x)$ of $X$ at $x$ is the fiber of the $(n-1)$-coskeleton-projection over $x$, hence the pullback

$$\begin{array}{ccc}{E}_n(X,x)& \to & X\\ \downarrow & & \downarrow \\ *& \stackrel{x}{\to }& {\mathrm{cosk}}_{n-1}X\end{array}.$$\array{ E_n(X,x) &\to& X \\ \downarrow && \downarrow \\ * &\stackrel{x}{\to}& cosk_{n-1}X } \,.

By the skeleton/coskeleton adjunction $({\mathrm{sk}}_{n-1}⊣{\mathrm{cosk}}_{n-1})$ 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 ${\mathrm{sSet}}_{(n-1)}\hookrightarrow {\mathrm{sSet}}^{*/}$ of $n$-fold reduced simplicial sets (those with a single $k$-simplex for all $k\le n-1$) into all pointed simplicial sets is a coreflective subcategory with coreflector being forming of the $n$th Eilenberg subcomplex

$${\mathrm{sSet}}^{*/}\stackrel{\stackrel{}{\hookleftarrow }}{\underset{E_n(-,*)}{\to }}{\mathrm{sSet}}_{n-1}.$$sSet^{*/} \stackrel{\overset{}{\hookleftarrow}}{\underset{E_n(-,*)}{\to}} sSet_{n-1} \,.

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

So if $(*\to X)\in {\mathrm{sSet}}^{*/}$ such that $X\in \mathrm{sSet}$ 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

$$\mathrm{\infty }{\mathrm{Grpd}}_{\ge (n-1)}^{*/}\hookrightarrow \mathrm{\infty }{\mathrm{Grpd}}^{*/}.$$\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)