One of my aims in writing this paper was to make the close connections between the theories of subfactors and of tensor categories completely explicit with the goal of furthering the interaction between them. To be sure, those connections shine though in the work of authors like V.F.R. Jones and A. Ocneanu, but they had been really appreciated only by very few workers in the field of subfactors, like H. Wenzl and S. Yamagami. I doubt that my paper has done much to change this state of affairs. But it seems to have been quite effective at convincing people in other fields that they have something to learn from subfactor theory. (After all, Jones and Ocneanu have been very successful in using subfactor theory in obtaining important results in low dimensional topology. Jones received the Fields medal for this work.) Perhaps even this interpretation is too optimistic, but in any case I managed to show that a certain mathematical construction abstracted from subfactor theory can be generalized to the setting of category theory. By stripping away inessential technicalities I opened the subject to workers from other fields, in particular category and representation theorists. For both groups this new construction is of considerable interest.

I cannot claim to have discovered a new result in the theory of subfactors, but I have realized that the structures present there can be generalized and applied elsewhere, like in low dimensional topology, avoiding the technicalities of subfactors. I wouldn't go as far as speaking of a new methodology, but rather of a more abstract (and therefore simpler!) way of looking at certain things.

The main significance of the paper is that it provides further evidence for the connection between two fields of mathematics, subfactor theory and category theory. (All of subfactor theory and much of category theory are concerned with a generalized version of Galois theory, which is the problem of classifying how one instance of some mathematical structure, e.g., a field, can sit inside another such instance.) This connection makes it possible to transfer already existent ideas, methods, and results from one of the fields to the other and vice versa, as I already did in a sequel to the paper. More importantly, the improved communication between the two fields should accelerate future progress in subfactor theory and perhaps in category theory. Progress in this direction would also be to the benefit of areas where these theories are applied, like topological and conformal quantum field theory and, ultimately perhaps, string theory.