Maybe one can approach this from the other side. Knowing electromagnetism today, what is the most complete and simple framework for modelling it?
The set of equations from Maxwell's book are to me a heterogeneous presentation. If there is a category of electromagnetic objects or perhaps some other pure mathematics framework that synthesises everything together, then I would call that thing the whole.
Sure. You can start from relativity and show that for example the antisymmetric Faraday tensor (P in the link [1] below) can get you a lot of phenomena (not all the way, accelerating charges are hairy, even for Feynman, see the link [2])
To note, the reference to finding the relativistic invariant from the tensor in the first link goes back to the first edition of Landau and Lifshitz. The problem was removed from later editions because only a masochist wants to find invariants of 4x4 matrices by hand.
The set of equations from Maxwell's book are to me a heterogeneous presentation. If there is a category of electromagnetic objects or perhaps some other pure mathematics framework that synthesises everything together, then I would call that thing the whole.