In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are classes, gauge classes, and Grothendieck class. Could someone further explain these three terms?

Note that I am a physicist and am approaching this from the perspective of trying to rigorously understand anyons. So if you could provide an answer in the terminology of, say, Steven H. Simon's Topological Quantum: Lecture Notes and Proto-Book, that would be ideal.

In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are monoidal classes, gauge classes, and Grothendieck class. Could someone further explain these three terms?

Note that I am a physicist and am approaching this from the perspective of trying to rigorously understand anyons. So if you could provide an answer in the terminology of, say, Steven H. Simon's Topological Quantum: Lecture Notes and Proto-Book, that would be ideal.

In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are classes, gauge classes, and Grothendieck class. Could someone further explain these three terms?

Note that I am a physicist and am approaching this from the perspective of trying to rigorously understand anyons. So if you could provide an answer in the terminology of, say, this book, that would be ideal.

In the comments under this post, three notions of equivalence classes of unitary modular tensor categories are brought up. They are classes, gauge classes, and Grothendieck class. Could someone further explain these three terms?

Note that I am a physicist and am approaching this from the perspective of trying to rigorously understand anyons. So if you could provide an answer in the terminology of, say, Steven H. Simon's Topological Quantum: Lecture Notes and Proto-Book, that would be ideal.