Levels in the toposes of simplicial sets and cubical sets

TL;DR

This paper investigates the relationships between levels, skeleta, and coskeleta in various toposes like simplicial, cubical, and globular sets, providing new insights into their structural properties relevant to homotopy and higher category theory.

Contribution

It characterizes when n-skeletal objects imply k-coskeletal objects in different toposes, extending understanding of their hierarchical complexity.

Findings

n-skeletal cyclic sets are (2n+1)-coskeletal

n-skeletal implies (n+1)-coskeletal for globular sets

n-skeletal implies (2n-1)-coskeletal for simplicial 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 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.