  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 Fr\"olicher-Richardson operator algebra.
  The Lorentz covariance is the cornerstone of physical theory.
Observer-dependence within relativity groupoid, and the Lorentz-covariance
(within the Lorentz relativity group), are different concepts. Laws of Physics
could be observer-free, rather than Lorentz-invariant.
