arXiv 1907.025742019-07-04math.COmath.CT

图与超图理论的函子方法

Functorial Approach to Graph and Hypergraph Theory

Martin Schmidt

备注：: Thesis based on arXiv:1807.09345, arXiv:1807.09348

摘要：

利用范畴语法和范畴语义为范畴图和超图理论提供了一种新的研究方法。对于每个幺半群$M$和一个集合$X$上的动作，有一个相关的$(X,M)$ -图的预层拓扑，其中每个对象可以被解释为一个广义均匀超图，其中每个边都有基数$\#X$事件顶点（包括多重性），并且幺半群通知边拥有什么类型的凝聚力。$(X,M)$ -图的一个显著特征是存在不固定边。我们证明了，如果一个图或一致超图的指数和有效的等价关系存在的范畴中，不固定的边缘是一个必要的特征。将语法（$(X,M)$ -图论）与语义（$(X,M)$ -图的范畴）分离的主要优点是能够在任何共完备范畴中解释该理论。这个interpetation函子，然后产生一个神经实现的adjunction，并允许我们转移结构之间的类别的$(X,M)$ -图和接受的共完全范畴。

英文摘要：

We provide a new approach to categorical graph and hypergraph theory by using categorical syntax and semantics. For each monoid $M$ and action on a set $X$, there is an associated presheaf topos of $(X,M)$-graphs where each object can be interpreted as a generalized uniform hypergraph where each edge has cardinality $\#X$ incident vertices (including multiplicity) and where the monoid informs what type of cohesivity the edges possess. One distinguishing feature of $(X,M)$-graphs is the presence of unfixed edges. We prove that unfixed edges are a necessary feature of a category of graphs or uniform hypergraphs if one wants exponentials and effective equivalence relations to exist in the category. The main advantage of separating syntax (the $(X,M)$-graph theories) from semantics (the categories of $(X,M)$-graphs) is the ability to interpret the theory in any cocomplete category. This interpetation functor then yields a nerve-realization adjunction and allows us to transfer structure between the category of $(X,M)$-graphs and the receptive cocomplete category.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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