nLab simplicial skeleton

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

Definition

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

$$tr_n : sSet = [\Delta^{op}, Set] \to [\Delta_{\leq n}^{op},Set] = sSet_{\leq n}$$

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

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

$$sk_n \;\colon\; sSet_{\leq n} \to sSet$$

called the $n$-skeleton

and a fully faithful right adjoint, given by right Kan extension

$$cosk_n \;\colon\; sSet_{\leq n} \to sSet$$

called the $n$-coskeleton.

$$( sk_n \dashv tr_n \dashv cosk_n) \;\; \colon \;\; sSet_{\leq n} \stackrel{\overset{sk_n}{\longrightarrow}}{\stackrel{\overset{tr_n}{\longleftarrow}}{\underset{cosk_n}{\longrightarrow}}} sSet \,.$$

The $n$-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree $n$. Conversely, the $n$-coskeleton produces a simplicial set having a simplex of degree $m \gt n$ whenever there is a compatible family of $m$-faces.

Write

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

and

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

for the composite functors. Often by slight abuse of notation we suppress the boldface and just write $sk_n : sSet \to sSet$ and $cosk_n : sSet \to sSet$.

these in turn form an adjunction

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

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

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

Simplicial sets isomorphic to objects in the image of $cosk_n$ are called $n$-coskeletal simplicial sets.

Properties

General

Proposition

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

• $X$ is $n$-coskeletal;

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

• the map

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

  is a bijection for all $k \gt n$

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

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

Remark

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

Compatibility with Kan conditions

Proposition

The coskeleton operations $\mathbf{cosk}_n$ preserve Kan complexes.

More generally, $\mathbf{cosk}_n$ preserves those Kan fibrations between Kan complexes whose codomains have trivial homotopy group $\pi_n$.

(Dwyer & Kan 1984, p. 141 (4 of 9), proofs are spelled out by Low 2013, Deflorin 2019, Lemma 10.12)

Truncation and Postnikov towers

Proposition

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

$$X \longrightarrow \mathbf{cosk}_n(X)$$

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

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

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

is a Postnikov tower for $X$.

See also the discussion in Dwyer & Kan 1984, p. 140, 141.

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

Example

(coskeletality of simplicial nerves of categories)
The simplicial nerve of a category (i.e. of a 1-category) is a 2-coskeletal simplicial set (this Prop.): The unique filler of the boundary of an $n \geq 3$-simplex encodes the associativity-condition on $n$-tuples of composable morphisms.

Of course there is more to a category than its associativity condition, and hence the converse fails: Not every 2-coskeletal simplicial set is the nerve of a category. For example the boundary of the 2-simplex, $\partial \Delta^2$, is 2-coskeletal but not the nerve of a category, since it is missing a composition of the edges $0 \to 1 \to 2$, namely it is missing a filler of this inner horn.

In fact, a simplicial set is the nerve of a category iff it has unique inner $n$-horn-fillers for $n \geq 2$ (e.g. this Prop.). But 2-coskeletality already implies that all $k \geq 4$-horns have unique filler (first uniquely fill the missing $k-1$-face then the interior $k$)-cell. Together this implies that:

A simplicial set is the nerve of a category iff

1. it is 2-coskeletal,

2. all inner 2- and 3-horns have unique fillers (encoding composition and associativity).

Similarly for groupoids (by this Prop.):

A simplicial set is the nerve of a groupoid iff

1. it is 2-coskeletal,

2. all 2- and 3-horns have unique fillers.

For better or worse, such a simplicial set has at times also been called a 1-hypergroupoid, pointing to the fact that this is the first non-trivial stage in a pattern that recognizes $n+1$-coskeletal Kan complexes with unique horn fillers as models for $n$-groupoids

Notice that a Kan complex which is 2-coskeletal but with possibly non-unique 2-horn fillers is still a homotopy 1-type and may still be called a 1-groupoid in the sense of homotopy theory, but need not be the nerve of a groupoid. It may be thought of as a bigroupoid (2-hypergroupoid) which happens to be just a homotopy 1-type.

Accordingly, essentially by definition:

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

Also:

• 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

• adjoint modality

• hypergroupoid

References

• Peter May, Section II.8 of: Simplicial objects in algebraic topology, The University of Chicago Press 1967 (djvu, ISBN:9780226511818)

• William Dwyer, Dan Kan, Section 1.2 (vi) of: An obstruction theory for diagrams of simplicial sets, Indagationes Mathematicae (Proceedings) 87 2 (1984) 139-146 [doi:10.1016/1385-7258(84)90015-5]

• Paul Goerss, J. F. Jardine, Section VI.3 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4)

Also:

• Zhen Lin, Math.SE:a/597990

• Gian Deflorin, Section 10.2 of: The Homotopy Hypothesis, Zurich 2019 (2019, pdf)

The level of a topos-structure of simplicial (co-)skeleta is discussed in

• C. Kennett, E. Riehl, M. Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets , JPAA 215 no.5 (2011) pp.949-961. (arXiv:1003.5944)