On reachability categories, persistence, and commuting algebras of quivers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912108593020928 |
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| author | Caputi, Luigi Riihimäki, Henri |
| author_facet | Caputi, Luigi Riihimäki, Henri |
| contents | For a finite quiver $Q$, we study the reachability category $\mathbf{Reach}_Q$. We investigate the properties of $\mathbf{Reach}_Q$ from both a categorical and a topological viewpoint. In particular, we compare $\mathbf{Reach}_Q$ with $\mathbf{Path}_Q$, the category freely generated by $Q$. As a first application, we study the category algebra of $\mathbf{Reach}_Q$, which is isomorphic to the commuting algebra of $Q$. As a consequence, we recover, in a categorical framework, previous results obtained by Green and Schroll; we show that the commuting algebra of $Q$ is Morita equivalent to the incidence algebra of a poset, the reachability poset. We further show that commuting algebras are Morita equivalent if and only if the reachability posets are isomorphic. As a second application, we define persistent Hochschild homology of quivers via reachability categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15388 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On reachability categories, persistence, and commuting algebras of quivers Caputi, Luigi Riihimäki, Henri Rings and Algebras Combinatorics Category Theory For a finite quiver $Q$, we study the reachability category $\mathbf{Reach}_Q$. We investigate the properties of $\mathbf{Reach}_Q$ from both a categorical and a topological viewpoint. In particular, we compare $\mathbf{Reach}_Q$ with $\mathbf{Path}_Q$, the category freely generated by $Q$. As a first application, we study the category algebra of $\mathbf{Reach}_Q$, which is isomorphic to the commuting algebra of $Q$. As a consequence, we recover, in a categorical framework, previous results obtained by Green and Schroll; we show that the commuting algebra of $Q$ is Morita equivalent to the incidence algebra of a poset, the reachability poset. We further show that commuting algebras are Morita equivalent if and only if the reachability posets are isomorphic. As a second application, we define persistent Hochschild homology of quivers via reachability categories. |
| title | On reachability categories, persistence, and commuting algebras of quivers |
| topic | Rings and Algebras Combinatorics Category Theory |
| url | https://arxiv.org/abs/2306.15388 |