Category Theory: Symmetry Group of Comma-propagation Transformations
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917703646707712 |
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| author | Majkic, Zoran |
| author_facet | Majkic, Zoran |
| contents | In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced by Lawvere in the context of the interdefinability of the universal concepts of category theory. The basic idea is the elevation of arrows of one category $\textbf{C}$ to objects in another. Given a category (as a "geometric object") $\textbf{C}$ we can consider its properties (the universal categorial commutative diagrams) preserved under actions of a comma-propagation operation $\{\}$ in the infinite hierarchy of its arrow-categories (n-dimensional levels, such that for any $n\geq 1$, $\textbf{C}_{n+1} = {\textbf{C}_n}$, with $\textbf{C}_1 =\textbf{C}$) and on the functors (and their natural transformations) between such n-dimensional levels, which is a phenomena of a general categorial symmetry under a categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Category Theory: Symmetry Group of Comma-propagation Transformations Majkic, Zoran General Mathematics In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced by Lawvere in the context of the interdefinability of the universal concepts of category theory. The basic idea is the elevation of arrows of one category $\textbf{C}$ to objects in another. Given a category (as a "geometric object") $\textbf{C}$ we can consider its properties (the universal categorial commutative diagrams) preserved under actions of a comma-propagation operation $\{\}$ in the infinite hierarchy of its arrow-categories (n-dimensional levels, such that for any $n\geq 1$, $\textbf{C}_{n+1} = {\textbf{C}_n}$, with $\textbf{C}_1 =\textbf{C}$) and on the functors (and their natural transformations) between such n-dimensional levels, which is a phenomena of a general categorial symmetry under a categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations. |
| title | Category Theory: Symmetry Group of Comma-propagation Transformations |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2406.16884 |