Associative n-categories
Associative n-categories
复制标题
关联 n 类别
DOI：
--
复制
发表时间：
2018
期刊：
影响因子：
--
通讯作者：
C. Dorn
中科院分区：
文献类型：
--
作者：
C. Dorn
我们定义了新的高范畴的完全组合模型。我们的定义是基于更高范畴与“有向空间”的联系。有向空间是在流形图上局部模拟的，流形图是n立方体的分层，使得地层横切于n立方体的旗叶。本文的第一部分建立了流形图的一个组合模型，称为奇异n-立方体。在第二部分中，我们应用这个模型来建立我们的更高范畴的概念。
奇数n立方体是空间的“有向三角剖分”，并分解成一系列的子空间或层。单数n立方体可以自然地组织成两类。第一种方法用于奇异(n+1)立方体的归纳定义，其态射本身就是丛。第二种，它的态射是“开放的”基变化，承认(epi，mono)因式分解系统。单态将称为立方体的嵌入。满射将被称为折叠，并描述三角剖分如何被粗化。每个立方体都有一个独特的最粗三角剖分，称为其法线形式。范式的存在使得(组合表示的)流形图的等式关系是可判定的。
作为所得到的流形图的组合框架的主要应用，我们给出了更高范畴的各种概念的代数定义。即定义了结合n-范畴，给出了结合n-范畴，给出了结合n-群胚。这三个概念都将有严格的单位和结合子；唯一的弱相干性是同伦，但我们开发了一种机制来恢复弱n-范畴的通常相干性数据，例如结合子和五角体及其更高的类似物。这将引发一种猜想，即结合更高范畴的理论与其完全弱的对应理论是等价的。
We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of the n-cube such that strata are transversal to the flag foliation of the n-cube. The first part of this thesis develops a combinatorial model for manifold diagrams called singular n-cubes. In the second part we apply this model to build our notions of higher categories.
Singular n-cubes are "directed triangulations" of space together with a decomposition into a collection of subspaces or strata. Singular n-cubes can be naturally organised into two categories. The first, whose morphisms are bundles themselves, is used for the inductive definition of singular (n+1)-cubes. The second, whose morphisms are "open" base changes, admits an (epi,mono) factorisation system. Monomorphisms will be called embeddings of cubes. Epimorphisms will be called collapses and describe how triangulations can be coarsened. Each cube has a unique coarsest triangulation called its normal form. The existence of normal forms makes the equality relation of (combinatorially represented) manifold diagrams decidable.
As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. Namely, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. All three notions will have strict units and associators; the only weak coherences are homotopies, but we develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This will motivate the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.