Metacategory and Geometric Algebras (2403.19689v1)

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.