In the topos of simplicial sets, it makes sense to ask the following question about a given natural number $n$: what is the minimum value $m$ such that $n$-skeletality implies $m$-coskeletality? This is an instance of the Aufhebung relation in the sense of Lawvere, who introduced this notion for an arbitrary Grothendieck topos $\mathcal{E}$ in place of $\mathbf{sSet}$, and levels/essential subtopoi in place of dimensions.
We compute this Aufhebung relation for the topos of symmetric simplicial sets. In particular, we show that it is given by $2l-1$ for the level labelled by $l\geq 3$, which coincides with the previously known case of simplicial sets. This result provides a solution to the fourth of the seven open problems in topos theory posed by Lawvere in 2009.
Solution to Lawvere's first problem: a Grothendieck topos that has proper class many quotient topoi
Graduate School of Mathematical Sciences, The University of Tokyo（东京大学大学院数理科学研究科）；Graduate School of Mathematical Sciences, University of Tokyo（东京大学大学院数理科学研究科）
In this article, we introduce certain kinds of computable reduction games with imperfect information. One can view such a game as an extension of the notion of Turing reduction, and generalized Weihrauch reduction as well. Based on the work by Lee and van Oosten, we utilize these games for providing a concrete description of the lattice of the Lawvere-Tierney topologies on the effective topos (equivalently, the subtoposes of the effective topos preordered by geometric inclusion). As an application, for instance, we show that there exists no minimal Lawvere-Tierney topology which is strictly ab…
This paper deals with the notion of weak Lawvere-Tierney topology on a topos. Our motivation to study such a notion is based on the observation that the composition of two Lawvere-Tierney topologies is no longer idempotent, when seen as a closure operator. For a given topos $\mathcal{E}$, in this paper we investigate some properties of this notion. Among other things, it is shown that the set of all weak Lawvere-Tierney topologies on $\mathcal{E}$ constitutes a complete residuated lattice provided that $\mathcal{E}$ is (co)complete. Furthermore, when the weak Lawvere-Tierney topology on $\math…
We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the definitions of a topological and uniform space. We use these embeddings to rewrite the notions of completeness, precompactness, compactness, Cauchy sequence, and equicontinuity in the language of category theory, which we hope might be of use in formalisation of mathematics and tame topology. We formulate some arising open questions.
We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g. continuity, limit of a map or a sequence along a filter, various notions of equicontinuity and uniform convergence of a sequence of functions; completeness and compactness; in algebraic topology, locally trivial bundles as a direct product after base-change and geometric realisation as a space of discontinuous paths. In model theory, we observe that indiscern…
Lawvere has observed that certain 'gros' toposes in algebraic geometry suggest the existence of an 'infinitesimal level', closely related to finite-dimensional local algebras. Motivated by this observation we propose an elementary definition of level ε associated to a local geometric morphism, establish some relevant basic properties suggested by geometric intuition, and give concrete descriptions of the level ε determined by several pre-cohesive geometric morphisms.
在本文中，我们考虑单纯的家庭，也就是说，单纯对象索引的单纯集。我们开发了一种方法来构建家庭超覆盖加细的覆盖家庭的基础上的概念，我们在这里介绍的\emph{n-spans}。在[The fundamental progroupoid of a general topos，Journal of Pure and Applied Algebra 212（2008）]中，我们在topos中引入了\emph{covering projection}的概念。它们是局部常数对象，满足一个附加条件，当拓扑是局部连通的时，该条件在所有局部常数对象中都是有效的，并发展了一般拓扑的基本广群理论。在这里，我们表明，覆盖投影可以得到从一个家庭的集合上的单纯集的下降数据构造的对象。我们构造了一个广群$\nn{G}_\cc{H}$，使得被$\cc{H}$平凡化的覆盖投影范畴$\cc{G}_\cc{H}$是它的分类拓扑.这确定了一个原群$\{\cc{G} \cc{H}\}_\cc{H}$和一个progroupoid $\{\nn{G}_\cc{H}\}_\cc{H}$，适合于由超覆盖的过滤偏序集索引。然后，我们表明，这progroupoid分类torsors。这种构造在局部连通拓扑的情况下也是新颖的，表明局部连通拓扑中的局部常数对象是通过从集合族上的下降数据下降而构造的。区分局部连通topoi的显著特征是，pro…
A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$-simplices given by the simplicial maps $Δ[n] \times K \to X$. We prove that $X^K$ is non-singular whenever $X$ is non-singular. It follows that non-singular simplicial sets form a cartesian closed category with all limits and colimits, but it is not a topos.
