A Categorical Approach to Möbius Inversion via Derived Functors
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909379419176960 |
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| author | Elchesen, Alex Patel, Amit |
| author_facet | Elchesen, Alex Patel, Amit |
| contents | We develop a cohomological approach to Möbius inversion using derived functors in the enriched categorical setting. For a poset $P$ and a closed symmetric monoidal abelian category $\mathcal{C}$, we define Möbius cohomology as the derived functors of an enriched hom functor on the category of $P$-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical Möbius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota's Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04362 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Categorical Approach to Möbius Inversion via Derived Functors Elchesen, Alex Patel, Amit Algebraic Topology Combinatorics Category Theory K-Theory and Homology We develop a cohomological approach to Möbius inversion using derived functors in the enriched categorical setting. For a poset $P$ and a closed symmetric monoidal abelian category $\mathcal{C}$, we define Möbius cohomology as the derived functors of an enriched hom functor on the category of $P$-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical Möbius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota's Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra. |
| title | A Categorical Approach to Möbius Inversion via Derived Functors |
| topic | Algebraic Topology Combinatorics Category Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2411.04362 |