On Maximal, Universal and Complete Extensions of Yang-Mills-Type Theories

In this paper we continue the program on the classification of extensions of the Standard Model of Particle Physics started in arXiv:2007.01660. We propose four complementary questions to be considered when trying to classify any class of extensions of a fixed Yang-Mills-type theory $S^G$: existence problem, obstruction problem, maximality problem and universality problem. We prove that all these problems admits a purely categorical characterization internal to the category of extensions of $S^G$. Using this we show that maximality and universality are dense properties, meaning that if they are not satisfied in a class $\mathcal{E}(S^G;\hat{G})$, then they are in their "one-point compactification" $\mathcal{E}(S^G;\hat{G})\cup \hat{S}$ by a specific trivial extension $\hat{S}$. We prove that, by means of assuming the Axiom of Choice, one can get another maximality theorem, now independent of the trivial extension $\hat{S}$. We consider the class of almost coherent extensions, i.e, complete, injective and of pullback-type, and we show that for it the existence and obstruction problems have a complete solution. Using again the Axiom of Choice, we prove that this class of extensions satisfies the hypothesis of the second maximality theorem.