Paths and cylinders
This entry is about the notion of (co)skeleta of simplicial sets. For the notion of skeleton of a category see 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 left adjoint, given by left Kan extension
called the -skeleton
and a 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 .
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:
on the unit of the adjunction is an isomorphism;
is a bijection for all
for and every morphism from the boundary of the -simplex there exists a unique filler
Truncation and Postnikov towers
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 on p. 140, 141 of DwKan1984.
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.
Standard textbook references are
A classical article that amplifies the connection of the coskeleton operation to Postnikov towers is
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)