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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?

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    $\begingroup$ I do not know much about phisics... but "semisimple with only finitely many isomorphism classes of simple objects" means that you can choose a finite se of simple(=indecomposable) objects $\{C_1,\dots,C_n\}$ such that, any other object $C$ in the category, can be decomposed as a coproduct $C\cong C_1^{(I_1)}\coprod\cdots\coprod C_n^{(I_n)}$, where the $I_i$ are some, uniquely determined by $C$, finite index sets. Also, as a consequence, you can write morphisms in that category in a nice matricial form. $\endgroup$ Dec 18, 2020 at 13:48
  • $\begingroup$ @SimoneVirili The isomorphism part is what I don't yet understand, thanks for the rest of the comment! So isomorphisms of simple objects would be some functors from object1 to object2 and vice versa, that are inverses of each other? So if I have F: C_1 times C_2 will that give me a unique product of simple objects in the same class? $\endgroup$ Dec 19, 2020 at 1:52
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    $\begingroup$ "Finitely many isomorphism classes" just means that, up to isomorphism, you have just finitely many simples. To be completely precise, you can put it this way: take the class $S$ of all simple objects in your category, then there is an equivance class on $S$, defined by saying that two elements of $S$ are equivalent if and only if they are isomorphic. "Finitely many isomorphism classes" means that, in $S$, there are finitely many equivalence classes with respect to this equivalence relation. $\endgroup$ Dec 19, 2020 at 9:32
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    $\begingroup$ For example, take a field $\mathbb F$, and let $\mathcal C$ be the category of vector spaces over $\mathbb F$. Then, since all simple objects in $\mathcal C$ are isomorphic to each other (they are all isomorphic to the one-dimensional vector space $\mathbb F$), you have just one isomorphism class of simple objects. $\endgroup$ Dec 19, 2020 at 9:35
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    $\begingroup$ @SimoneVirili that example helped a lot in understanding the definition! Thanks! $\endgroup$ Dec 19, 2020 at 9:58

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