On topological representation theory from quivers

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 $At^Γ(T,f)$.