Levels in the toposes of simplicial sets and cubical sets

Abstract

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 This paper, which subsumes earlier unpublished work of some of the authors, answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical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 ( 2 n − 1 ) -coskeletal for simplicial sets and 2 n -coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n -skeletal cyclic sets are necessarily ( 2 n + 1 ) -coskeletal.