Levels in the toposes of simplicial sets and cubical sets

Carolyn Kennett, Emily Riehl*, Michael Roy, Michael Zaks

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    2 Citations (Scopus)


    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 This paper, which subsumes earlier unpublished work of some of the authors, answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical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 (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.

    Original languageEnglish
    Pages (from-to)949-961
    Number of pages13
    JournalJournal of Pure and Applied Algebra
    Issue number5
    Publication statusPublished - May 2011


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