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nLab simplicial skeleton

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

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see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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

Contents

Definition

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.

Write

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

and

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.

Properties

General

Proposition

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] }
Remark

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

Proposition

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

Proposition

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.

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 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:

Also:

References

Also:

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 December 3, 2024 at 16:41:47. See the history of this page for a list of all contributions to it.