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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 the simplex category write for its full subcategory on the objects . The inclusion induces a truncation functor
that takes a simplicial set and restricts it to its degrees .
This functor has a fully faithful left adjoint, given by left Kan extension
called the -skeleton
and a fully faithful right adjoint, given by right Kan extension
called the -coskeleton.
The -skeleton produces a simplicial set that is freely filled with degenerate simplices above degree . Conversely, the -coskeleton produces a simplicial set having a simplex of degree whenever there is a compatible family of -faces.
Write
and
for the composite functors. Often by slight abuse of notation we suppress the boldface and just write and .
these in turn form an adjunction
So the -coskeleton of a simplicial set is given by the formula
Simplicial sets isomorphic to objects in the image of are called -coskeletal simplicial sets.
For sSet, the following are equivalent:
is -coskeletal;
on the adjunction unit is an isomorphism;
the map
is a bijection for all
for and every morphism from the boundary of the -simplex there exists a unique filler
So in particular if is an -coskeletal Kan complex, all its simplicial homotopy groups above degree are trivial.
The coskeleton operations preserve Kan complexes.
More generally, preserves those Kan fibrations between Kan complexes whose codomains have trivial homotopy group .
(Dwyer & Kan 1984, p. 141 (4 of 9), proofs are spelled out by Low 2013, Deflorin 2019, Lemma 10.12)
For each , the unit of the adjunction
induces an isomorphism on all simplicial homotopy groups in degree .
It follows from the above that for a Kan complex, the sequence
is a Postnikov tower for .
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 -simplex encodes the associativity-condition on -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, , is 2-coskeletal but not the nerve of a category, since it is missing a composition of the edges , 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 -horn-fillers for (e.g. this Prop.). But 2-coskeletality already implies that all -horns have unique filler (first uniquely fill the missing -face then the interior )-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 -coskeletal Kan complexes with unique horn fillers as models for -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 December 3, 2024 at 16:41:47. See the history of this page for a list of all contributions to it.