Normed modules and the categorification of integrations, series expansions, and differentiations
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| Format: | Preprint |
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2024
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| _version_ | 1866916713324347392 |
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| author | Liu, Yu-Zhe Liu, Shengda Huang, Zhaoyong Zhou, Panyue |
| author_facet | Liu, Yu-Zhe Liu, Shengda Huang, Zhaoyong Zhou, Panyue |
| contents | We explore the assignment of norms to $\mathitΛ$-modules over a finite-dimensional algebra $\mathitΛ$, resulting in the establishment of normed $\mathitΛ$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathitΛ$-module $N$ limited by a special element $v_N\in N$ and a special $\mathitΛ$-homomorphism $δ_N: N^{\oplus 2^{\dim\mathitΛ}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathitΛ$-homomorphism $θ: N\to M$ such that $θ(v_N) = v_M$ and $θδ_N = δ_Mθ^{\oplus 2^{\dim\mathitΛ}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. By examining the objects and morphisms in these categories. We establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_02777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Normed modules and the categorification of integrations, series expansions, and differentiations Liu, Yu-Zhe Liu, Shengda Huang, Zhaoyong Zhou, Panyue Representation Theory Category Theory Functional Analysis 16G10, 46B99, 46M40 We explore the assignment of norms to $\mathitΛ$-modules over a finite-dimensional algebra $\mathitΛ$, resulting in the establishment of normed $\mathitΛ$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathitΛ$-module $N$ limited by a special element $v_N\in N$ and a special $\mathitΛ$-homomorphism $δ_N: N^{\oplus 2^{\dim\mathitΛ}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathitΛ$-homomorphism $θ: N\to M$ such that $θ(v_N) = v_M$ and $θδ_N = δ_Mθ^{\oplus 2^{\dim\mathitΛ}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. By examining the objects and morphisms in these categories. We establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$. |
| title | Normed modules and the categorification of integrations, series expansions, and differentiations |
| topic | Representation Theory Category Theory Functional Analysis 16G10, 46B99, 46M40 |
| url | https://arxiv.org/abs/2405.02777 |