  The essential subtoposes of a fixed topos form a complete lattice, which
gives rise to the notion of a level in a topos. In the familiar example of
simplicial sets, levels coincide with dimensions and give rise to the usual
notions of n-skeletal and n-coskeletal simplicial sets. In addition to the
obvious ordering, the levels provide a stricter means of comparing the
complexity of objects, which is determined by the answer to the following
question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This
paper answers this question for several toposes of interest to homotopy theory
and higher category theory: simplicial sets, cubical sets, and reflexive
globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the
other two examples the situation is considerably more complicated: n-skeletal
implies (2n-1)-coskeletal for simplicial sets and 2n-coskeletal for cubical
sets, but nothing stronger. In a discussion of further applications, we prove
that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.
