On Topological Representation Theory from Quivers

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发表日期：

2024年5月21日

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Fang Li,
Zhihao Wang,
Jie Wu,
Bin Yu

摘要

In this work, we introduce 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. Firstly, we investigate the relation between the category of topological representations and that of linear representations of a quiver via $P(\varGamma )$ $P (Γ)$ - $\mathcal {TOP}^o$ ${TOP}^{o}$ and $k\varGamma $ $k Γ$ -Mod, concerning (positively) graded or vertex (positively) graded modules. Secondly, we discuss the homological theory of topological representations of quivers via the $\varGamma $ $Γ$ -limit functor $lim ^{\varGamma }$ $l i m^{Γ}$ , and use it to define the homology groups of topological representations of quivers via $H _n$ $H_{n}$ . It is found that some properties of a quiver can be read from homology groups. Thirdly, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in ${\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma $ $Top - Rep Γ$ and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we obtain the functor $At^{\varGamma }$ $A t^{Γ}$ from ${\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma $ $Top - Rep Γ$ to ${\textbf {Top}}$ $Top$ and show that $At^{\varGamma }$ $A t^{Γ}$ preserves homotopy equivalence between morphisms. The relationship between the homotopy groups of a top-representation (T, f) and the homotopy groups of $At^{\varGamma }(T,f)$ $A t^{Γ} (T, f)$ is also established.