ncatlab defines a fusion category as "A fusion category is a rigid semisimple linear (Vect-enriched) monoidal category (“tensor category”), with only finitely many isomorphism classes of simple objects, such that the endomorphisms of the unit object form just the ground field k.".
How does this decidedly mathematical definition relate to a physical system say, hosting anyons in a 2 dimensional quantum hall system? Or really any other physical example you can think of? I wanted to understand the reasoning behind, and the properties of the mathematical definitions of a fusion category.
So far I understand that it is a category with a tensor product structure, that is you can build objects and morphisms by tensoring previously defined objects and morphisms. This would make sense when you have say a lattice and spins on each link, and you can take the system as a tensor product of all spins.
rigid and linear all make sense, because you want the objects (spins in this particular case) to have a dual (given by daggering it), linear (since you should be able to add the spin directions on each vertex to get a composite spin). This is intuitive because you can treat the spins on a link as vectors, which already have all these properties. How does semisimple fit in?
I don't understand isomorphism onwards, what exactly does that imply? Am I missing any grave properties that I should be knowing that make this category? Have I made the right connections to the physical perspective, or am I thinking of it incorrectly?