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Metacategory and Geometric Algebras

Zoran Majkic

Abstract

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, can not be defined by using well-know Grothendeick's approach with discrete ringed spaces. So, we can consider any category 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 paths. Based on this approach, we define the Cat-vector space V valid for all categories with noncommutative (and partial) addition operation for the vectors, and their inner product. For the categories wher4e 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.

Metacategory and Geometric Algebras

Abstract

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, can not be defined by using well-know Grothendeick's approach with discrete ringed spaces. So, we can consider any category 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 paths. Based on this approach, we define the Cat-vector space V valid for all categories with noncommutative (and partial) addition operation for the vectors, and their inner product. For the categories wher4e 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.
Paper Structure (6 sections, 2 theorems, 20 equations)

This paper contains 6 sections, 2 theorems, 20 equations.

Table of Contents

  1. A Short Introduction to Geometric Algebras
  2. Introduction to Metacategory and its Geometric Space
  3. A General Definition of Geometric Metacategory Space
  4. Categorial Vector Space
  5. Category Theory and Geometry
  6. Conclusion

Key Result

Corollary 1

Vector base $B$ of the Cat-vector space $V$ is normal base with $\|e_i\| = 1$ for each base vector $e_i \in B$. However, it is not generally an orthonormal base because we can have, of some category, two base vectors $e_i, e_j \in B$ which are not mutually orthogonal.

Theorems & Definitions (7)

  • definition 1
  • definition 2
  • definition 3
  • Corollary 1
  • definition 4
  • definition 5
  • Corollary 2