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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.10837 |
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| _version_ | 1866908956156231680 |
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| author | Brüstle, Thomas Desrochers, Justin Leblanc, Samuel |
| author_facet | Brüstle, Thomas Desrochers, Justin Leblanc, Samuel |
| contents | Let $\mathcal{C}$ be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory $\mathcal{J}$ is both initial and final, then the restriction of any $\mathcal{C}$-module along $\mathcal{J}$ preserves the generalized rank-or equivalently, the multiplicity of the ``entire" interval modules for $\mathcal{C}$ and $\mathcal{J}$. Conversely, we prove that this property characterizes initial and final embeddings when both $\mathcal{C}$ and $\mathcal{J}$ are posets satisfying certain mild constraints and the embedding is full. For $\mathcal{C}$ a poset under these conditions, we describe the minimal full subposet whose embedding is initial or final. This generalizes an observation made by Dey and Lesnick. We also extend a result of Kinser on the generalized rank invariant to small categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10837 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Rank via Minimal Subposet Brüstle, Thomas Desrochers, Justin Leblanc, Samuel Representation Theory Category Theory 16G20, 18A30, 18A25 (Primary) 55N31 (Secondary) Let $\mathcal{C}$ be a small, connected category with finite hom-sets. We show that if the embedding of a connected subcategory $\mathcal{J}$ is both initial and final, then the restriction of any $\mathcal{C}$-module along $\mathcal{J}$ preserves the generalized rank-or equivalently, the multiplicity of the ``entire" interval modules for $\mathcal{C}$ and $\mathcal{J}$. Conversely, we prove that this property characterizes initial and final embeddings when both $\mathcal{C}$ and $\mathcal{J}$ are posets satisfying certain mild constraints and the embedding is full. For $\mathcal{C}$ a poset under these conditions, we describe the minimal full subposet whose embedding is initial or final. This generalizes an observation made by Dey and Lesnick. We also extend a result of Kinser on the generalized rank invariant to small categories. |
| title | Generalized Rank via Minimal Subposet |
| topic | Representation Theory Category Theory 16G20, 18A30, 18A25 (Primary) 55N31 (Secondary) |
| url | https://arxiv.org/abs/2510.10837 |