nLab
simplicial skeleton

This entry is about the notion of (co)skeleta of simplicial sets. For the notion of skeleton of a category see skeleton.

Definition

For $\Delta$ the simplex category write ${\Delta }_{\le n}$ for its full subcategory on the objects $[0],[1],\cdots ,[n]$. The inclusion $\Delta {\mid }_{\le n}\hookrightarrow \Delta$ induces a truncation functor

$${\mathrm{tr}}_n:\mathrm{sSet}=[{\Delta }^{\mathrm{op}},\mathrm{Set}]\to [{\Delta }_{\le n},\mathrm{Set}]$$tr_n : sSet = [\Delta^{op}, Set] \to [\Delta_{\leq n},Set]

that takes a simplicial set and restricts it to its degrees $\le n$.

This functor has a left adjoint, given by left Kan extension

$${\mathrm{sk}}_n:[{\Delta }_{\le n},\mathrm{Set}]\to \mathrm{SSet}$$sk_n : [\Delta_{\leq n},Set] \to SSet

called the $n$-skeleton

and a right adjoint, given by right Kan extension

$${\mathrm{cosk}}_n:[{\Delta }_{\le n},\mathrm{Set}]\to \mathrm{SSet}$$cosk_n : [\Delta_{\leq n},Set] \to SSet

called the $n$-coskeleton.

$$({\mathrm{sk}}_n⊣{\mathrm{tr}}_n⊣{\mathrm{cosk}}_n):{\mathrm{sSet}}_{\le n}\stackrel{\stackrel{{\mathrm{sk}}_n}{\to }}{\stackrel{\stackrel{{\mathrm{tr}}_n}{\leftarrow }}{\underset{{\mathrm{cosk}}_n}{\to }}}\mathrm{sSet}.$$( sk_n \dashv tr_n \dashv cosk_n) \;\; : \;\; sSet_{\leq n} \stackrel{\overset{sk_n}{\to}}{\stackrel{\overset{tr_n}{\leftarrow}}{\underset{cosk_n}{\to}}} sSet \,.

The $n$-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree $n$.

Write

$${\mathrm{sk}}_n:={\mathrm{sk}}_n\circ {\mathrm{tr}}_n:\mathrm{sSet}\to \mathrm{sSet}$$\mathbf{sk}_n := sk_n \circ tr_n: sSet \to sSet

and

$${\mathrm{cosk}}_n:={\mathrm{cosk}}_n\circ {\mathrm{tr}}_n:\mathrm{sSet}\to \mathrm{sSet}$$\mathbf{cosk}_n := cosk_n \circ tr_n: sSet \to sSet

for the composite functors. Often by slight abuse of notation we suppress the boldface and just write ${\mathrm{sk}}_n:\mathrm{sSet}\to \mathrm{sSet}$ and ${\mathrm{cosk}}_n:\mathrm{sSet}\to \mathrm{sSet}$.

these in turn form an adjunction

$$({\mathrm{sk}}_n⊣{\mathrm{cosk}}_n):\mathrm{sSet}\overleftarrow{\to }\mathrm{sSet}.$$( \mathbf{sk}_n \dashv \mathbf{cosk}_n) \;\; : \;\; sSet \stackrel{\leftarrow}{\to} sSet \,.

So the $k$-coskeleton of a simplicial set $X$ is given by the formula

$${\mathrm{cosk}}_{k}X:[n]\mapsto {\mathrm{Hom}}_{\mathrm{sSet}}({\mathrm{sk}}_k\Delta [n],X).$$\mathbf{cosk}_k X : [n] \mapsto Hom_{sSet}(\mathbf{sk}_k \Delta[n], X) \,.

Simplicial sets isomorphic to objects in the image of ${\mathrm{cosk}}_n$ are called coskeletal simplicial sets.

Properties

General

Proposition

For $X\in$ sSet, the following are equivalent:

• $X$ is $n$-coskeletal;

• on $X$ the unit $X\to {\mathrm{cosk}}_n(X)$ of the adjunction is an isomorphism;

• the map

  $$X_k=\mathrm{Hom}(\Delta [k],X)\stackrel{{\mathrm{tr}}_n}{\to }\mathrm{Hom}({\mathrm{tr}}_n(\Delta [k]),{\mathrm{tr}}_n(X))$$X_k = Hom(\Delta[k], X) \stackrel{tr_n}{\to} Hom(tr_n(\Delta[k]), tr_n(X))

  is a bijection for all $k>n$

• for $k>n$ and every morphism $\partial \Delta [k]\to X$ from the boundary of the $k$-simplex there exists a unique filler $\Delta [k]\to X$

  $$\begin{array}{ccc}\partial \Delta [k]& \to & X\\ \downarrow & \nearrow \\ \Delta [k]\end{array}$$\array{ \partial \Delta[k] &\to& X \\ \downarrow & \nearrow \\ \Delta[k] }

Remark

So in particular if $X$ is an $n$-coskeletal Kan complex, all its simplicial homotopy groups above degree $(n-1)$ are trivial.

Truncation and Postnikov towers

Proposition

For each $n\in \mathbb{N}$, the unit of the adjunction

$$X\to {\mathrm{cosk}}_n(X)$$X \to \mathbf{cosk}_n(X)

induces an isomorphism on all simplicial homotopy groups in degree $<n$.

It follows from the above that for $X$ a Kan complex, the sequence

$$X=\underleftarrow{\lim }{\mathrm{cosk}}_{n}X\to \cdots \to {\mathrm{cosk}}_{n+1}X\to {\mathrm{cosk}}_{n}X\to \cdots \to *$$X = \underset{\leftarrow}{\lim}\, cosk_n X \to \cdots \to cosk_{n+1} X \to cosk_{n} X \to \cdots \to *

is a Postnikov tower for $X$.

See also the discussion on p. 140, 141 of DwKan1984.

For the interpretation of this in terms of (n,1)-toposes inside the (∞,1)-topos ∞Grpd see n-truncated object in an (∞,1)-category, example In ∞Grpd and Top.

Examples

• The nerve of a category is a 2-coskeletal simplicial set.

• A Kan complex that is $(n+1)$-coskeletal is equivalent to (the nerve of) an n-groupoid.

• A 0-coskeletal simplicial set $X$ is (-1)-truncated and hence either empty or a contractible Kan complex , $X\stackrel{\simeq }{\to }*$ that is the nerve $X=N(C)$ of a groupoid $C$ that has a equivalence of categories $C\simeq *$.

Related concepts

• Postnikov tower, Whitehead tower

• Eilenberg subcomplex

References

Standard textbook references are

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

• Paul Goerss and Rick Jardine, 1999, Simplicial homotopy theory, number 174 in Progress in Mathematics, Birkhäuser. (ps)

A classical article that amplifies the connection of the coskeleton operation to Postnikov towers is

• William Dwyer, Dan Kan, An obstruction theory for diagrams of simplicial sets (pdf)