Eilenberg subcomplex



For XX a simplicial set, for x:Δ[0]Xx : \Delta[0] \to X a point in XX, and for nn \in \mathbb{N}, the nnth Eilenberg subcomplex E n(X,x)E_n(X,x) of XX at xx is the fiber of the (n1)(n-1)-coskeleton-projection over xx, hence the pullback

E n(X,x) X * x cosk n1X. \array{ E_n(X,x) &\to& X \\ \downarrow && \downarrow \\ * &\stackrel{x}{\to}& cosk_{n-1}X } \,.

By the skeleton/coskeleton adjunction (sk n1cosk n1)(sk_{n-1} \dashv cosk_{n-1}) the nnth Eilenberg subcomplex is the subobject of XX consisting of those simplices whose (n1)(n-1)-skeleton is constant on the point xx.


Restriction to Kan complexes

If XX is a Kan complex , then so is E n(X,x)E_n(X,x) for all nn \in \mathbb{N} and xX 0x \in X_0.

Relation to nn-connected objects

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

See (May, theorem 8.4).

Relation to pointed nn-connected objects

The inclusion sSet (n1)sSet */sSet_{(n-1)} \hookrightarrow sSet^{*/} of nn-fold reduced simplicial set (those with a single kk-simplex for all kn1k \leq n-1) into all pointed simplicial sets is a coreflective subcategory with coreflector being forming of the nnth Eilenberg subcomplex

sSet */E n(,*)sSet n1. sSet^{*/} \underoverset {\underset{E_n(-,*)}{\longrightarrow}} {\overset{}{\hookleftarrow}} {\bot} sSet_{n-1} \,.

the counit of this adjunction is the defining inclusion E n(X,*)XE_n(X,*) \to X.

So if (*X)sSet */(* \to X) \in sSet^{*/} such that XsSetX \in sSet is a Kan complex and (n-1)-connected, then the counit E n(X,*)XE_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

Grpd (n1) */Grpd */. \infty Grpd_{\geq (n-1)}^{*/} \hookrightarrow \infty Grpd^{*/} \,.


Around def. 8.3 in

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