Standard Model only uses U(1), SU(2) and SU(3). Note that Lie groups do not come equipped with a metric. A yet more complicated, yet more accurate and geometrically enlightening, approach is to understand that the gauge covariant derivative is (exactly) the same thing as the exterior covariant derivative on a section of an associated bundle for the principal fiber bundle of the gauge theory; and, for the case of spinors, the associated bundle would be a spin bundle of the spin structure. Although conceptually the same, this approach uses a very different set of notation, and requires a far more advanced background in multiple areas of differential geometry. The final step in the geometrization of gauge invariance is to recognize that, in quantum theory, one needs only to compare neighboring fibers of the principal fiber bundle, and that the fibers themselves provide a superfluous extra description. This leads to the idea of modding out the gauge group to obtain the gauge groupoid as the closest description of the gauge connection in quantum field theory. For ordinary Lie algebras, the gauge covariant derivative on the space symmetries (those of the pseudo-Riemannian manifold and general relativity) cannot be intertwined with the internal gauge symmetries; that is, metric geometry and affine geometry are necessarily distinct mathematical subjects: this is the content of the Coleman–Mandula theorem. However, a premise of this theorem is violated by the Lie superalgebras (which are not Lie algebras!) thus offering hope that a single unified symmetry can describe both spatial and internal symmetries: this is the foundation of supersymmetry. The more mathematical approach uses an index-free notation, emphasizing the geometric and algebraic structure of the gauge theory and its relationship to Lie algebras and Riemannian manifolds; for example, treating gauge covariance as equivariance on fibers of a fiber bundle. The index notation used in physics makes it far more convenient for practical calculations, although it makes the overall geometric structure of the theory more opaque. The physics approach also has a pedagogical advantage: the general structure of a gauge theory can be exposed after a minimal background in multivariate calculus, whereas the geometric approach requires a large investment of time in the general theory of differential geometry, Riemannian manifolds, Lie algebras, representations of Lie algebras and principle bundles before a general understanding can be developed. In more advanced discussions, both notations are commonly intermixed. This article attempts to follow