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Paths and cylinders

Homotopy groups

Basic facts


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



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

tr n:sSet=[Δ op,Set][Δ n op,Set]=sSet n 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 n\leq n.

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

sk n:sSet nsSet sk_n \;\colon\; sSet_{\leq n} \to sSet

called the nn-skeleton

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

cosk n:sSet nsSet cosk_n \;\colon\; sSet_{\leq n} \to sSet

called the nn-coskeleton.

(sk ntr ncosk n):sSet ncosk ntr nsk nsSet. ( 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 nn-skeleton produces a simplicial set that is freely filled with degenerate simplices above degree nn. Conversely, the nn-coskeleton produces a simplicial set having a simplex of degree m>nm \gt n whenever there is a compatible family of mm-faces.


sk n:=sk ntr n:sSetsSet \mathbf{sk}_n := sk_n \circ tr_n: sSet \to sSet


cosk n:=cosk ntr n:sSetsSet \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:sSetsSetsk_n : sSet \to sSet and cosk n:sSetsSetcosk_n : sSet \to sSet.

these in turn form an adjunction

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

So the kk-coskeleton of a simplicial set XX is given by the formula

cosk kX:[n]Hom sSet(sk kΔ[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 cosk ncosk_n are called nn-coskeletal simplicial sets.




For XX \in sSet, the following are equivalent:

  • XX is nn-coskeletal;

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

  • the map

    X k=Hom(Δ[k],X)tr nHom(tr n(Δ[k]),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>nk \gt n

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

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

So in particular if XX is an nn-coskeletal Kan complex, all its simplicial homotopy groups above degree (n1)(n-1) are trivial.

Compatibility with Kan conditions


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

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


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

Xcosk n(X) X \longrightarrow \mathbf{cosk}_n(X)

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

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

X=limcosk nXcosk n+1Xcosk nX* 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 XX.

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.



(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 3 n \geq 3 -simplex encodes the associativity-condition on n 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, Δ 2\partial \Delta^2, is 2-coskeletal but not the nerve of a category, since it is missing a composition of the edges 0120 \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 n -horn-fillers for n2n \geq 2 (e.g. this Prop.). But 2-coskeletality already implies that all k4k \geq 4-horns have unique filler (first uniquely fill the missing k1k-1-face then the interior kk)-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+1n+1-coskeletal Kan complexes with unique horn fillers as models for n 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:




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)

Last revised on February 5, 2024 at 10:50:21. See the history of this page for a list of all contributions to it.