nLab
simplicial 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.

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

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

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

$$({\mathrm{cosk}}_n⊣{\mathrm{sk}}_n):\mathrm{sSet}\overleftarrow{\to }\mathrm{sSet}.$$(\mathbf{cosk}_n \dashv \mathbf{sk}_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

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] }

Examples

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

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

• A 0-coskeletal simplicial set $X$ is 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 $C\simeq *$.

References

• P. G. Goerss and J. F. Jardine, 1999, Simplicial Homotopy Theory, number 174 in Progress in Mathematics, Birkhauser. (ps)