Relativity Groupoid, Instead of Relativity Group

Minkowski in 1908 used space-like binary velocity-field of a medium, relative to an observer. Hestenes in 1974 introduced, within a Clifford algebra, an axiomatic binary relative velocity as a Minkowski bivector. We propose consider binary relative velocity as a traceless nilpotent endomorphism in an operator algebra. Any concept of a binary axiomatic relative velocity made possible the replacement of the Lorentz relativity group by the relativity groupoid. The relativity groupoid is a category of massive bodies in mutual relative motions, where a binary relative velocity is interpreted as a categorical morphism with the associative addition. This associative addition is to be contrasted with non-associative addition of (ternary) relative velocities in isometric special relativity (loop structure). We consider an algebra of many time-plus-space splits, as an operator algebra generated by idempotents. The kinematics of relativity groupoid is ruled by associative Frobenius operator algebra, whereas the dynamics of categorical relativity needs the non-associative FrölicherRichardson operator algebra. The Lorentz covariance is the cornerstone of physical theory. Observerdependence within relativity groupoid, and the Lorentz-covariance