因果网范畴
Causal-net category
AI总结：
本文引入因果网范畴$\boldsymbol{\text{Cau}}$，证明其态射可分解为六类不可分解态射，研究了其中多种偏序关系与广义子式，为因果网及图论研究提供了自然的范畴论框架。
摘要：
因果网是一种有限无环有向图。本文引入了一个记为$\boldsymbol{\text{Cau}}$的范畴，称为因果网范畴，其对象为因果网，两个因果网之间的态射是它们的路径范畴之间的函子。$\boldsymbol{\text{Cau}}$范畴实际上是“因果网上的自由范畴”单子的Kleisli范畴。首先，我们阐述了研究$\boldsymbol{\text{Cau}}$的动机，并说明了其在因果网凝聚框架下的应用。我们证明恰好存在六类不可分解态射，对应于幺半范畴图形演算的六种约定。其次，我们研究了$\boldsymbol{\text{Cau}}$中若干态射的合成闭类，这些类刻画了因果网之间有意义的偏序关系，例如粗粒化、合并、收缩、浸入子式、拓扑子式等，并证明了若干有用的分解定理。第三，我们为子式理论引入了一个范畴框架，并利用它研究了$\boldsymbol{\text{Cau}}$中若干类型的广义子式。此外，我们证明了一个基本定理：$\boldsymbol{\text{Cau}}$中的任意态射都是六类不可分解态射的合成，并表明着色和恰当子式的概念可分别被理解为$\boldsymbol{\text{Cau}}$中特殊的最小商和子商。基于这些结果，我们得出结论：$\boldsymbol{\text{Cau}}$是研究因果网的自然框架，$\boldsymbol{\text{Cau}}$的理论应为图论的范畴论理解提供新的视角。
英文摘要：
A causal-net is a finite acyclic directed graph. In this paper, we introduce a category, denoted by $\mathbf{Cau}$ and called causal-net category, whose objects are causal-nets and morphisms between two causal-nets are the functors between their path categories. The category $\mathbf{Cau}$ is in fact the Kleisli category of the "free category on a causal-net" monad. Firstly, we motivate the study of $\mathbf{Cau}$ and illustrate its application in the framework of causal-net condensation. We show that there are exactly six types of indecomposable morphisms, which correspond to six conventions of graphical calculi for monoidal categories. Secondly, we study several composition-closed classes of morphisms in $\mathbf{Cau}$, which characterize interesting partial orders among causal-nets, such as coarse-graining, merging, contraction, immersion-minor, topological minor, etc., and prove several useful decomposition theorems. Thirdly, we introduce a categorical framework for minor theory and use it to study several types of generalized minors in $\mathbf{Cau}$. In addition, we prove a fundamental theorem that any morphism in $\mathbf{Cau}$ is a composition of the six types of indecomposable morphisms, and show that the notions of coloring and exact minor can be understood as special kinds of minimal-quotient and sub-quotient in $\mathbf{Cau}$, respectively. Base on these results, we conclude that $\mathbf{Cau}$ is a natural setting for studying causal-nets, and the theory of $\mathbf{Cau}$ should shed new light on the category-theoretic understanding of graph theory.
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