组合胞腔复形：对偶、重构与因果配边
Combinatorial Cell Complexes: Duality, reconstruction and causal cobordisms
摘要：
本文提出了一个基于组合胞元复形的框架，其胞元被简单地定义为有限个顶点的集合。一个cc的单元格服从四个公理，涉及一个为每个单元格分配秩（或维度）的秩函数。我们的框架专注于类CC承认一个包含反转对偶映射。我们引入一个组合概念的配边，使我们能够挑出一个范畴，其态射的配边具有因果结构。我们的目标是提供一种方法来寻找一个组合的概念，量子场论具有一个内置的对偶操作的基础空间上，而不依赖于任何流形结构。导言包括与量子引力相关的理论和数学物理领域的链接，并激发了我们的框架。首先，我们引入CC和一类称为闭CC的空边界CC上的对偶映射。然后，我们专注于重建一定类别的CC从他们的细胞秩2和更低的问题。这样的CC是特别是无边界的单形复形，它们的重建是使用离散的连接概念来实现的。我们的下一个主要结果将我们在闭cc上定义的对偶映射推广到一类有边界的cc上。一个重要的副产品的研究，这种扩展的对偶映射是组合概念的协边在这项工作中使用。我们还介绍了一个一般的概念细分的CC通过一个地图称为减少，以及减少称为崩溃的对偶概念。这两种类型的映射使用称为切片序列的映射序列来表征称为切片的某些cc的结构。切片是我们定义因果配边的基本构建块，切片序列的对偶定义了配边的组成，为我们提供了一个范畴，其态射是因果配边。
英文摘要：
This thesis proposes a framework based on a notion of combinatorial cell complex (cc) whose cells are defined simply as finite sets of vertices. The cells of a cc are subject to four axioms involving a rank function that assigns a rank (or a dimension) to each cell. Our framework focuses on classes of cc admitting an inclusion-reversing duality map. We introduce a combinatorial notion of cobordism that allows us to single out a category whose morphisms are cobordisms having a causal structure. Our aim is to offer an approach to look for a combinatorial notion of quantum field theory having a built-in duality operation acting on the underlying space and not relying on any manifold structure. The introduction includes links with fields in Theoretical and Mathematical Physics related to Quantum Gravity and motivating our framework. We start by introducing cc and the duality map on a class of cc with empty boundary called closed cc. We then focus on the problem of reconstructing a certain class of cc from their cells of rank 2 and lower. Such cc are in particular duals to simplicial complexes with no boundary and their reconstruction is realized using a discrete notion of connection. Our next main result extends the duality map we defined on closed cc to a class of cc with boundary. An important by-product of the study of this extended duality map is the combinatorial notion of cobordism used in this work. We also introduce a general notion of subdivision of a cc via a map called reduction, as well as the dual notion of reduction called collapse. These two types of map characterize the structure of certain cc called slices, using sequences of maps called slice sequences. Slices are the basic building blocs of our definition of causal cobordisms and the dual of a slice sequence defines the composition of cobordisms, providing us with a category whose morphisms are causal cobordisms.
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