Levels in the toposes of simplicial sets and cubical sets

arXiv:1003.5944v1 [math.CT] 30 Mar 2010 LEVELS IN THE TOPOSES OF SIMPLICIAL SETS AND CUBICAL SETS CAROLYN KENNETT, EMILY RIEHL, MICHAEL ROY, MICHAEL ZAKS 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 obvi- ous 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 This paper, which subsumes earlier unpublished work of some of the authors, answers this ques- tion for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical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 situ- ation 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. Contents 1. Introduction 1 2. Aufhebung of cubical sets 3 3. Aufhebung of simplicial sets 9 References 19 1. Introduction Consider a geometric morphism between toposes B and A, i.e., a functor B →A with a finite limit preserving left adjoint. If the right adjoint is fully faithful, we say that B is a subtopos of A. If the left adjoint itself has a left adjoint, then we say B is an essential subtopos of A, in which case we have a diagram: A i∗⊥ ⊥ / B i∗ Y i!  The right adjoint inclusion of B into A is a geometric morphism, which we think of as the sheaf inclusion of the essential subtopos. By contrast, the left adjoint inclusion, sometimes called “essentiality,” is not typically a geometric morphism, Date: October 25, 2018. 1 2 CAROLYN KENNETT, EMILY RIEHL, MICHAEL ROY, MICHAEL ZAKS though in examples this is often the more natural way to think about objects of the subtopos in the context of the larger topos. Kelly and Lawvere show that the essential subtoposes of a given topos form a complete lattice [KL89]. In light of this result, each such subtopos B is referred to as a level of A. For each level B, i!i∗defines a comonad skB and i∗i∗defines a monad coskB on A such that skB is left adjoint to coskB. For example, suppose A is the topos of presheaves on some small category ∆. Any fully faithful inclusion i : ∆′ ֒→∆induces functors Set∆op i∗ ⊥ ⊥ / Set(∆′)op i! | i∗ b where i∗is restriction and i! and i∗are left and right Kan extension. These functors exhibit Set∆′op as an essential subtopos of Set∆op. Up to isomorphism, the functor i∗is a common retraction of i! and i∗, which are both fully faithful. This situation has been called unity and identity of opposites [Law96], [Law91]. An object A of A is B-skeletal if A ∼= skBA; likewise A is B-coskeletal if A ∼= coskBA. A level B′ is lower than a level B if the skeletal and coskeletal inclusions of B′ into A factor through the skeletal and coskeletal inclusions, respectively, of B in A. In the above example, the category of presheaves on a full subcategory ∆′′ ֒→∆′ is lower than the category of presheaves on ∆′ . A level B′ is way below a level B if in addition its skeletal inclusion into A factors through the coskeletal inclusion of B in A, i.e., if B′-skeletal implies B-coskeletal. The smallest level B in the lattice of essential subtoposes of A for which this condition holds, if such a level exists, is called the Aufhebung of B′, terminology introduced by Lawvere in deference to Hegel [Law91]. In three toposes which have been important for the study of homotopy theory and higher category theory — simplicial sets [GZ67] [May92], cubical sets [Kan55], and reflexive globular sets [Str00] — levels coincide with dimensions: the category of presheaves on a small category is equivalent to the presheaves on its Cauchy completion. Up to splitting of idempotents, the distinct full subcategories of, e.g., the simplicial category ∆are the categories ∆n on objects [0], . . . , [n] for each natural number. Thus, dimensions classify the essential subtoposes of the category of simplicial sets; a similar proof works for the other examples. For these toposes, a level n is lower than a level k precisely when n ≤k, and the question of determining the Aufhebung of the level n can be stated more colloquially: when does n-skeletal imply k-coskeletal? Naively, one might hope that n-skeletal implies (n+1)-coskeletal, and for reflexive globular sets this is indeed the case, as was first observed by Roy [Roy97]. A reflexive globular set is a presheaf on the globe category G, with the natural numbers as objects and maps of the form σ, τ : n →n + 1 such that τσ = σσ and ττ = στ and ι : n + 1 →n such that ισ = id = ιτ. For reflexive

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