The categorical relativity is a groupoid category of massive bodies in mutual motions. The relative velocity is defined to be the basis-free and coordinatefree binary morphism. We are showing that coordinate-free unique definition of relative velocity in Galilean relativity becomes two different coordinate-free possibilities for relative simultaneity: binary velocity-morphism in categorical relativity, and ternary reciprocal-velocity in isometric special relativity. We are proving that the isometric Lorentz transformation needs at least threebody system. Observer-dependence and the Lorentz-covariance are different concepts.
The Poincare-Lorentz versus the Einstein-Minkowski interpretations of a formal structure of relativity are not the unique dichotomy. We propose to consider the concept of relative velocity as the primary concept with two possibilities: Voigt 1887 & Heaviside 1888, versus Einstein 1905. In categorical relativity the inverse relative-velocity-morphism v−1 is interior-observer-dependent, and not absolute as in the isometric exterior formulation where v−1 = −v.
In the framework of categorical relativity we consider coordinate-free transformations of adopted mathematical co-frames, and transformation of proper-times (clocks), including the transformation of the Einstein-Minkowski simultaneity. The categorical relativity does not predicts the lenght/rod material contraction, because this concept is not basis-free. The concept of simultaneity is basis-free and coordinate-free, and simultaneity in categorical relativity must be relative exactly in the same way as in the isometric special relativity.
As another example we consider the electric field registered by a moving observer: electrodynamics of moving bodies is different from the isometric special relativity with Lorentz transformations.
The kinematics of categorical relativity is ruled by Frobenius algebra, whereas the dynamics of categorical relativity needs the Fr?licher-Richardson algebra.
This work also review the mathematical and theoretical aspects of biological time-dilation and material length-contraction, with comparison with Langevin's interpretation in 1911.
