On Topological Representation Theory from Quivers

Abstract

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 \(P(\varGamma )\)-\(\mathcal {TOP}^o\) and \(k\varGamma \)-Mod, concerning (positively) graded or vertex (positively) graded modules. Secondly, we discuss the homological theory of topological representations of quivers via the \(\varGamma \)-limit functor \(lim ^{\varGamma }\), and use it to 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. Thirdly, we investigate the homotopy theory of topological representations of quivers. We define the homotopy equivalence between two morphisms in \({\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma \) and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we obtain the functor \(At^{\varGamma }\) from \({\textbf {Top}}\text{- }{} {\textbf {Rep}}\varGamma \) to \({\textbf {Top}}\) and show that \(At^{\varGamma }\) preserves homotopy equivalence between morphisms. The relationship between the homotopy groups of a top-representation (T, f) and the homotopy groups of \(At^{\varGamma }(T,f)\) is also established.