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.First, we investigate the relation between the category of topological representations and that of linear representations of a quiver via P (Γ)-T OP 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 Top−RepΓ 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 Top−RepΓ to 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 ).