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This entry is about the notion of (co)skeleta of simplicial sets. For the notion of skeleton of a category see at skeleton.
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
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
called the $n$-skeleton
and a fully faithful right adjoint, given by right Kan extension
called the $n$-coskeleton.
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
and
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
So the $k$-coskeleton of a simplicial set $X$ is given by the formula
Simplicial sets isomorphic to objects in the image of $cosk_n$ are called $n$-coskeletal simplicial sets.
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
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 $\Delta[k] \to X$
So in particular if $X$ is an $n$-coskeletal Kan complex, all its simplicial homotopy groups above degree $(n-1)$ are trivial.
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)
For each $n \in \mathbb{N}$, the unit of the adjunction
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
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.
(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
it is 2-coskeletal,
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
it is 2-coskeletal,
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:
Also:
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:
The level of a topos-structure of simplicial (co-)skeleta is discussed in
Last revised on February 5, 2024 at 10:50:21. See the history of this page for a list of all contributions to it.