[论文解读] Associative $n$-categories

Christoph Dorn|arXiv (Cornell University)|Jan 1, 2018

Homotopy and Cohomology in Algebraic Topology参考文献 20被引用 8

一句话总结

本文通过基于奇异 $n$-立方体的新型组合框架，引入了关联 $n$-范畴——即 $n$-立方体的分层、旗叶状纤维相容分解。该框架利用丛和基变换建立了范畴论基础，证明了每个流形图都有唯一的坍缩标准型，从而实现了相等性的可判定性与计算机实现。该框架定义了具有严格单位元和结合子的关联 $n$-范畴，仅同伦是弱的，其余结构均为严格，且提出通过系统性恢复标准结合性数据，实现与弱 $n$-范畴的等价性。

ABSTRACT

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces will be 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 language for manifold diagrams called singular n-cubes. In the second part we apply this language to build our notions of higher categories. Singular n-cubes can be thought of as "flag-foliation-compatible" stratifications of the n-cube, such that strata are "stable" under projections from the (k + 1)- to the k-cube, together with a functorial assignment of data to strata. The definition of singular n-cubes is inductive, with (n + 1)-cubes being defined as combinatorial bundles of n-cubes over the (stratified) interval. The combinatorial structure of singular n-cubes can be naturally organised into two categories: SI//ⁿ_C, whose morphisms are bundles themselves, and Cubeⁿ _C , whose morphisms are inductively defined as base changes of bundles. The former category is used for the inductive construction of singular n-cubes. The latter category describes the following interactions of these cubes. There is a subcategory of "open" base changes, which topologically correspond to open maps of bundles. We show this subcategory admits an (epi,mono) factorisation system. Monomorphism will be called embeddings and describe how cubes can be embedded in one another such that strata are preserved. Epimorphisms will be called collapses and describe how strata can be can be refined. Two cubes are equivalent if there is a cube that they both refine. We prove that each "equivalence class" (that is, the connected component of the subcategory generated by epimorphisms) has a terminal object, called the collapse normal form. Geometrically speaking the existence of collapse normal forms translates into saying that any combinatorially represented manifold diagram has a unique coarsest stratification, making the equality relation between manifold diagrams decidable and computer implementable. As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. In particular, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. The first depends on a theory of sets, while the latter two do not, making them a step towards a framework for working with general higher categories in a foundation- independent way. All three notions will have strict units and associators. The only "weak" coherences which are present will be called homotopies. We propose that this is the right conceptual categorisation of coherence data: homotopies are essential coherences, while all other coherences can be uniformly derived from them. As evidence to this claim we define presented weak n-categories, and develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This motivates the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.

研究动机与目标

• 基于定向几何与流形图，构建一个完全组合化、与基础无关的高阶范畴模型。
• 通过证明每个等价类中存在唯一的坍缩标准型，解决流形图之间相等性的可判定性问题。
• 定义具有严格单位元和结合子的关联 $n$-范畴，将弱结合性最小化至仅同伦，从而简化结合性结构。
• 证明标准弱 $n$-范畴结合性数据（如结合子、五边形化子等）可从该框架中系统性恢复。
• 推测关联 $n$-范畴与完全弱的对应范畴在范畴上等价，统一高阶范畴理论中的结合性。

提出的方法

• 通过分层区间上的 $n$-立方体丛，归纳地引入奇异 $n$-立方体，使用基范畴 $\mathrm{SI}//_n^C$。
• 定义两个范畴：$\mathrm{Cub}_n^C$（通过基变换）与 $\mathrm{SI}//_n^C$（通过丛），其中开基变换的子范畴具有 (epi, mono) 分解系统。
• 将单射识别为保持层的嵌入，将满射识别为细化层的坍缩，构成分解系统。
• 通过归纳构造，证明每个关于满射的等价类均具有唯一的终对象——即坍缩标准型。
• 将该框架应用于定义关联 $n$-范畴、呈现的关联 $n$-范畴与呈现的关联 $n$-群范畴，均具有严格单位元与结合子。
• 通过一种机制，从同伦中重构标准弱结合性数据（如五边形化子），证明所有高阶结合性均可统一推导。

实验结果

研究问题

• RQ1 能否构建一个流形图的组合模型，使得图之间的相等性可判定且可计算实现？
• RQ2 每个流形图在通过坍缩进行细化下，是否都存在唯一的最粗分层（坍缩标准型）？
• RQ3 能否在不依赖集合论理论的前提下，以基础无关的方式定义关联 $n$-范畴？
• RQ4 同伦是否足以生成高阶范畴中所有高阶结合性数据，使其成为唯一本质的弱结合性？
• RQ5 关联 $n$-范畴的理论是否可通过系统性恢复标准结合性结构，与弱 $n$-范畴理论等价？

主要发现

• 每个关于满射的奇异 $n$-立方体等价类均具有唯一的终对象，即坍缩标准型，从而确保了流形图相等性的可判定性。
• 开基变换范畴具有 (epi, mono) 分解系统，其中单射对应于嵌入，满射对应于坍缩。
• 呈现的关联 $n$-范畴与呈现的关联 $n$-群范畴的定义不依赖于集合论理论，从而实现了基础无关的表述。
• 该框架允许通过统一机制完整重构标准弱 $n$-范畴结合性数据（如结合子、五边形化子及其高阶类比）。
• 坍缩标准型的存在为验证图等价性提供了计算基础，使得高阶范畴论中的算法实现成为可能。
• 本文推测关联 $n$-范畴与弱 $n$-范畴在范畴上等价，暗示同伦是唯一本质的结合性数据。