What Is Categorical Relativity?

The categorical relativity is a groupoid category of massive bodies in mutual motions. The relative velocity is defined to be the basis-free and coordinate-free binary morphism. In categorical relativity there is no need to distinguish separately the constant relative velocities (special relativity) from the variable accelerated velocities, hence the categorical relativity goes beyond boundary of the special relativity, and unify the kinematics of the special and the general relativity theories for curved spacetime. The definition of relative velocity as the categorical morphism is contrasted with the reciprocal-velocity defined by means of the Lorentz boost in isometric special relativity. Observer-dependence and the Lorentz-covariance are different concepts. During XX century the Lorentz covariance was the cornerstone of physical theory. We predict that Lorentz covariance as well as Lorentz invariance will dwindle in importance. Physics must be observer-free, rather than Lorentz-invariant. In this paper instead of the individual time-plus-space split, we consider the algebra of many relative time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of categorical relativity is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative Frölicher-Richardson operator algebra. In categorical relativity the inverse relative-velocity-morphism v is interiorobserver-dependent, and not absolute as in the isometric exterior formulation where