Categorial Geometry and Algebraic Topology
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918400570163200 |
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| author | Majkic, Zoran |
| author_facet | Majkic, Zoran |
| contents | In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation. However, the Metacategory space, valid for all categories, cannot be defined by using well-know Grothendeick's approach with discrete ringed spaces. So, we can consider any category $\textbf{C}$ as an abstract geometric object,
that is, a discrete space where the points are the objects of this category and arrows between objects as the oriented paths. Based on this approach, we define the Cat-arrows space $V$ valid for all categories with commutative (and associative) partial addition operation $\oplus$ for the vectors, and their inner product. For the categories where we define the norm ("length") of the vectors in $V$ we can define also the outer (wedge) product of the vectors in $V$ and we show that such Cat-algebra satisfies two fundamental properties of the Clifford geometric algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorial Geometry and Algebraic Topology Majkic, Zoran General Mathematics In Categorial Topology, given a category (as a "geometric object") we can consider its properties preserved under continuous action (a "deformation") of a comma-propagation operation. However, the Metacategory space, valid for all categories, cannot be defined by using well-know Grothendeick's approach with discrete ringed spaces. So, we can consider any category $\textbf{C}$ as an abstract geometric object, that is, a discrete space where the points are the objects of this category and arrows between objects as the oriented paths. Based on this approach, we define the Cat-arrows space $V$ valid for all categories with commutative (and associative) partial addition operation $\oplus$ for the vectors, and their inner product. For the categories where we define the norm ("length") of the vectors in $V$ we can define also the outer (wedge) product of the vectors in $V$ and we show that such Cat-algebra satisfies two fundamental properties of the Clifford geometric algebra. |
| title | Categorial Geometry and Algebraic Topology |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2403.19689 |