In a physics course, dimensions (such as energy, length, duration), are taken as given. There are then certain algebraic rules associated to them: e.g. we can add energy with energy, and multiply energy with length, but we can't add an energy with a length. Moreover, there is a finite list of fundamental dimensions and other dimensions are derived from them.
We can state this as: physical dimensions form a free abelian group under multiplication over a basis of fundamental dimensions (1). But it is not obvious to me why there is a "group of dimensions" associated to physics, and why it has the particular structure that it has (e.g. it could have been the case that all dimensions can be derived merely from "time" and "length", or some arbitrary other group structure).
Is there a way to derive from fundamental models of physics (preferably classical, non-relativistic), that there is this "group of dimensions", and that we can associate a dimension to every measurement, and that this group has the particular structure that it has?
By analogy, we can derive from every topological space $T$, the fundamental group $G_T$. Is there a similar way to derive from a fundamental physical model, the "group of physical dimensions"? (Note that I'm not asking for a canonical basis of the group).
1. (we can pinpoint different bases, such as the ISQ one: length, mass, time, electric current, thermodynamic temperature, amount of substance, luminous intensity)
There is no fundamental reason why you should have a basis of 7. It all depends on what you include as physics. If you want to include something else, like information, then you would probably add bit to your group. If you define physics as only classical mechanics, you can have a basis of 3 (time, length, mass). If you include electromagnetism, you must include something for charge or current. And then you can expand, depending on the field. To make my point, look at the definition of luminous intensity, one of the 7 that you cite. It is a wavelength weighted power, and the weighting function is related to the biology of the human eye. So if the eye would see all the wavelengths with the same efficiency, then the candela would be just a multiple of Watt/sr/m.