Eilenberg subcomplex

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

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 $x$.

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

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).

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

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

Around def. 8.3 in

Revised on February 28, 2017 16:02:25
by Urs Schreiber
(185.25.95.132)