# nLab simplicial skeleton

## Theorems

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

# Contents

## Definition

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

${\mathrm{tr}}_{n}:\mathrm{sSet}=\left[{\Delta }^{\mathrm{op}},\mathrm{Set}\right]\to \left[{\Delta }_{\le n},\mathrm{Set}\right]$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

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

called the $n$-skeleton

and a right adjoint, given by right Kan extension

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

called the $n$-coskeleton.

$\left({\mathrm{sk}}_{n}⊣{\mathrm{tr}}_{n}⊣{\mathrm{cosk}}_{n}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}{\mathrm{sSet}}_{\le n}\stackrel{\stackrel{{\mathrm{sk}}_{n}}{\to }}{\stackrel{\stackrel{{\mathrm{tr}}_{n}}{←}}{\underset{{\mathrm{cosk}}_{n}}{\to }}}\mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$( 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

$\left({\mathrm{sk}}_{n}⊣{\mathrm{cosk}}_{n}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\mathrm{sSet}\stackrel{←}{\to }\mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$( \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

${\mathrm{cosk}}_{k}X:\left[n\right]↦{\mathrm{Hom}}_{\mathrm{sSet}}\left({\mathrm{sk}}_{k}\Delta \left[n\right],X\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $X\to {\mathrm{cosk}}_{n}\left(X\right)$ of the adjunction is an isomorphism;

• the map

${X}_{k}=\mathrm{Hom}\left(\Delta \left[k\right],X\right)\stackrel{{\mathrm{tr}}_{n}}{\to }\mathrm{Hom}\left({\mathrm{tr}}_{n}\left(\Delta \left[k\right]\right),{\mathrm{tr}}_{n}\left(X\right)\right)$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 $\partial \Delta \left[k\right]\to X$ from the boundary of the $k$-simplex there exists a unique filler $\Delta \left[k\right]\to X$

$\begin{array}{ccc}\partial \Delta \left[k\right]& \to & X\\ ↓& ↗\\ \Delta \left[k\right]\end{array}$\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 $\left(n-1\right)$ are trivial.

### Truncation and Postnikov towers

###### Proposition

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

$X\to {\mathrm{cosk}}_{n}\left(X\right)$X \to \mathbf{cosk}_n(X)

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

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

$X=\underset{←}{\mathrm{lim}}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{cosk}}_{n}X\to \cdots \to {\mathrm{cosk}}_{n+1}X\to {\mathrm{cosk}}_{n}X\to \cdots \to *$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 $X$.

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 $\left(n+1\right)$-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 $X=N\left(C\right)$ of a groupoid $C$ that has a equivalence of categories $C\simeq *$.

## References

Standard textbook references are

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

Revised on April 29, 2013 21:42:07 by Urs Schreiber (89.204.138.79)