General Categorial Geometry and Algebraic Topology

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, 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 morphisms 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, based on partial operation of categorial composition of morphisms, their inner and outer products in 3D Cat-arrows space, like in 3D Clifford algebra. We provide a general definition of the norm ("weight") of the vectors in $V$, cumulative under the composition of the category morphisms. This norm assigned to the morphisms, non existing in metacategory theory, enriches each individual category with new semantical interpretation of the morphisms: they are "weighted" with a real number. So, this general transformation of metacategory theory into 3D Cat-arrows space of vectors is not only another example of applicability of 3D Clifford algebra, but the new semantic (of their "weight") and topological enrichment of the Metacategory theory with its abstract undefined primitive concept of morphisms.