Category Theory: Symmetry Group of Comma-propagation Transformations

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the comma categories and were introduced by Lawvere in the context of the interdefinability of the universal concepts of category theory. The basic idea is the elevation of arrows of one category C to objects in another. Given a category (as a "geometric object") C we can consider its properties (the universal categorial commutative diagrams) preserved under actions of a comma-propagation operation \\ in the infinite hierarchy of its arrow-categories (n-dimensional levels, such that for any n≥ 1, Cₙ₊₁ = Cₙ, with C₁ =C) and on the functors (and their natural transformations) between such n-dimensional levels, which is a phenomena of a general categorial symmetry under a categorial-symmetry group CS(Z) of all comma-propagation transformations.