论文标题
图表集和重写较弱的高等类别
Diagrammatic sets and rewriting in weak higher categories
论文作者
论文摘要
我们重新访问Kapranov和Voevodsky在组合粘贴图上建立的空间的概念，现在是高维重写的框架，也是弱欧米茄类别模型的基础。在第一部分中，我们详细阐述了Steiner的指示复合体理论作为组合基础。我们个性化的定向复合物的便捷类别，并相对于一个这样的类别发展了示意集的理论。我们研究了一个与图表集的内部等效性的概念，并将其单形为弱欧米茄类别的模型，这些图形集的每个可综合图都通过等效到单个单元格连接。然后，我们定义一个提供代数复合材料的半分布模型，并研究严格的欧米茄类别中的嵌入到该模型中。最后，我们证明了弱模型中无穷基团的同型假设的一种版本，并在Kapranov和Voevodsky的证明中表现出了特定的错误，该错误以前是间接驳斥的。
We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on Steiner's theory of directed complexes as a combinatorial foundation. We individuate convenient classes of directed complexes and develop the theory of diagrammatic sets relative to one such class. We study a notion of equivalence internal to a diagrammatic set, and single out as models of weak omega-categories those diagrammatic sets whose every composable diagram is connected by an equivalence to a single cell. We then define a semistrict model providing algebraic composites and study the embedding of strict omega-categories into this model. Finally, we prove a version of the homotopy hypothesis for the infinity-groupoids in the weak model, and exhibit a specific mistake in a proof by Kapranov and Voevodsky that had previously been refuted indirectly.
