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 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.