论文标题
关于颤抖的拓扑表示理论
On topological representation theory from quivers
论文作者
论文摘要
在这项工作中，我们将Quiver的{\ em拓扑表示}作为由拓扑空间及其由箭量确定的关系组成的系统。这样的设置在箭袋的拓扑表示和拓扑空间的图表之间提供了自然的联系。 首先，我们研究了拓扑表示类别与通过$ p（γ）$ - $ \ MATHCAL {top}^o $和$kγ$ -MOD，涉及（肯定）分级或顶点（正面）分级模块的关系。其次，我们通过$γ$ -limit $ lim^γ$讨论了Quivers拓扑表示的同源理论，并使用它来定义Quivers的拓扑表示的同源组。发现可以从同源性组中读取颤抖的某些特性。第三，我们研究了颤动的拓扑表示的同义理论。我们在$ \ textbf {topbf {top} \ mathrm { - } \ textbf {rep}γ$中定义了两种形态之间的同倍等效性，并表明并行同型公理也适用于基于同型等值的顶级占主张。 最后，我们主要从$ \ textbf {top} \ mathrm { - } \ textbf {prop}γ$到$ \ textbf {topbf {top} $获得函数$ at^γ$，并显示$ at^γ$保留了形态学之间的均值。该关系是在顶级代表$（t，f）$和$ at^γ（t，f）$的同型组之间建立的。
In this work, we introduce {\em topological representations of a quiver} as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological representations of a quiver and diagrams of topological spaces. First, we investigate the relation between the category of topological representations and that of linear representations of a quiver via $P(Γ)$-$\mathcal{TOP}^o$ and $kΓ$-Mod, concerning (positively) graded or vertex (positively) graded modules. Second, we discuss the homological theory of topological representations of quivers via $Γ$-limit $Lim^Γ$ and using it, define the homology groups of topological representations of quivers via $H_n$. It is found that some properties of a quiver can be read from homology groups. Third, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in $\textbf{Top}\mathrm{-}\textbf{Rep}Γ$ and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we mainly obtain the functor $At^Γ$ from $\textbf{Top}\mathrm{-}\textbf{Rep}Γ$ to $\textbf{Top}$ and show that $At^Γ$ preserves homotopy equivalence between morphisms. The relationship is established between the homotopy groups of a top-representation $(T,f)$ and the homotopy groups of $At^Γ(T,f)$.
