AI 中文总结
本文将纺织系统扩展至三维，建立其与3-图的关联，证明三维纺织同调群与对应3-图同调群一致，为3-图提供了纺织类框架。
AI 中文摘要
纺织系统最著名的应用是建模二维有限型移位。本文中，我们将离散代数与纺织系统关联，并为其提供一个群胚模型。当纺织系统是左消解的时，该代数与关联2-图的Kumjian–Pask代数一致。本文的主要目标是将纺织系统扩展到3维，使所得结构一方面能捕获所有三维有限型移位，另一方面能为3-图提供类似纺织的框架，扩展2-图与左消解纺织系统之间的已知联系。我们引入三维纺织系统的模型，并研究此类纺织系统与3-图之间的相互作用。特别地，我们的研究表明，从3-色图形成3-图所需的条件（包括三色路径上微妙的结合性条件），可通过纺织数据产生的简单拉回图来编码。我们还定义了三维纺织的同调群，并证明这些群与关联3-图的同调群一致，从而确立我们的构造在同调层面与3-图一致。
英文摘要
Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian--Pask algebra of the associated $2$-graph. The main objective of this paper is to extend textile systems to dimension $3$ so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for $3$-graphs extending the well-known connection between $2$-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and $3$-graphs. In particular, our investigation shows that the conditions required to form a $3$-graph from a $3$-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated $3$-graphs, thus establishing that our construction is homologically consistent with $3$-graphs.
CommentsThis project is yet to be completed. Comments are very much welcome on this early draft
