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Simplicial approach to path homology of quivers, marked categories, groups and algebras

Sergei O. Ivanov, Fedor Pavutnitskiy

Abstract

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.

Simplicial approach to path homology of quivers, marked categories, groups and algebras

Abstract

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.
Paper Structure (81 sections, 78 theorems, 332 equations)

This paper contains 81 sections, 78 theorems, 332 equations.

Table of Contents

  1. Introduction
  2. Weak cylinder functors
  3. Graded submodules of chain complexes
  4. Complexes ω and ψ
  5. Functorial properties of ω
  6. Homotopy invariance of ω and ψ
  7. Relation of ω and ψ to the tensor product
  8. DG-(co)algebras
  9. Duality over fields
  10. Quivers
  11. Definition of a quiver
  12. Paths in a quiver
  13. Box product of quivers
  14. Path objects
  15. Simplicial indexing category Δ
  16. ...and 66 more sections

Key Result

Proposition 2.1

Let $\mathpzc{C}$ and $\widetilde{\mathpzc{C}}$ be two categories with weak cylinder functors $({\sf cyl},i^0,i^1)$ and $(\widetilde{\sf cyl}, \tilde{i}^0,\tilde{i}^1).$ Assume that $F:\mathpzc{C} \to \widetilde{\mathpzc{C}}$ is a functor and there is a natural transformation $\varphi : \widetilde{\

Theorems & Definitions (178)

  • Proposition 2.1
  • proof
  • Remark 3.1
  • Proposition 3.2: cf. grbic2022aspects
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 168 more