(Co)Homology Theories for Oriented Algebras

Bustamante, Juan Carlos; Dionne, Julie; Smith, David Stanley

doi:10.1080/00927870802089892

Cited by 2 publications

(2 citation statements)

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“…By Theorem 3.6 and Theorem 3.8, we know that for a finite connected quiver Γ, there is an arrow function on it if and only if there is an arrow positive function on it. Since all the proofs in [4] about gradation were based on positive gradation, all conclusions about gradation in [4] can be equivalently replaced by the ones about positive gradation. Similarly, our results about positive gradation in this paper can be equivalently replaced by the ones about gradation.…”

Section: Relations Between Top-representations and Linear-representat...mentioning

confidence: 99%

“…In [6], for a quiver Γ, Henrik Holm and P. Jorgensen introduced a Γ-bundle associated to two chains. In [4], Bustamante J C, Dionne J and Smith D discussed homology theories for oriented algebras. In this section we hope to associate a quiver to chain complexes and use the homology groups of these chain complexes to reflect the quiver.…”

Section: Homology Groups Of Top-representationsmentioning

confidence: 99%

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On topological representation theory from quivers

Li,

Wang,

et al. 2020

Preprint

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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 ).

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Section: Relations Between Top-representations and Linear-representat...mentioning

confidence: 99%

Section: Homology Groups Of Top-representationsmentioning

confidence: 99%