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On Topological Representation Theory from Quivers

Fang Li , Zhihao Wang , Jie Wu , icon-email Bin Yu

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(Γ)-TOPo and kΓ-Mod, concerning (positively) graded or vertex (positively) graded modules. Secondly, we discuss the homological theory of topological representations of quivers via the Γ-limit functor limΓ, and use it to define the homology groups of topological representations of quivers via Hn. 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 Top- RepΓ and show that the parallel Homotopy Axiom also holds for top-representations based on the homotopy equivalence. Last, we obtain the functor AtΓ from Top- RepΓ to Top and show that AtΓ preserves homotopy equivalence between morphisms. The relationship between the homotopy groups of a top-representation (T,f) and the homotopy groups of AtΓ(T,f) is also established.

Viện Toán học, Viện Hàn lâm Khoa học và Công nghệ Việt Nam

Địa chỉ: Số 18 đường Hoàng Quốc Việt, quận Cầu Giấy, Hà Nội

Điện thoại: 024 37563474 - Fax: 024 37564303

Email: vientoan@math.ac.vn

Viện nghiên cứu cao cấp về toán Hội Toán học Mỹ MathScinet Thư viện số (VAST) Arxiv
Lịch hoạt động chung Lịch sử dụng phòng, hội trường, hội thảo Ảnh hoạt động
Giới thiệu Đăng nhập Đăng ký tài khoản Quên mật khẩu Chính sách bảo mật
Messages2TimelineExceptionsViews8RouteQueries108Models431MailsGateSessionRequest
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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      Metadata
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