Category theory formulation of the Racah-Wigner calculus

The Racah-Wigner calculus transcribes canonically into category theory. Diagrams play an important part in computations within the resulting category. These diagrams are unlike any diagrams that are presently used for group theoretic calculations. The category diagram structure simplifies and clarifies typically complicated manipulations of the calculus. Category theory diagrams illustrate the construction of coupling and recoupling coefficients. Natural isomorphisms of the category define various types of recoupling coefficients. Fundamental relationships within the category imply the Biedenharn-Elliott Sum Rule and the Racah Backcoupling relationship.