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Consider the two statements:

  1. "Any unitary fusion category can be realised as a category of endomorphisms on a hyperfinite von Neumann algebra", as stated in 1506.03546 page 4. The above paper refers to (I think) Theorem 7.6 of this paper. In the next paragraph of 1506.03546, they say that this is true for non-unitary fusion categories as well.

  2. Not every unitary fusion category is strict, e.g. the unitary fusion category associated to any finite group G and a nontrivial 3-cocycle of G.

Now, I was previously under the impression that any category of endomorphisms is strict, but the two statements above show that this is wrong. What is a simple example that illustrates that a category of endomorphisms can have nontrivial associator?

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Every monoidal category is monoidally equivalent to a strict one. This is MacLane’s famous coherence theorem.

What the example $\mathrm{Vec}(G, \omega)$ shows is that you can’t always make it simultaneously strict and skeletal. But you can do either separately if you like.

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  • $\begingroup$ Thank you! Given that any fusion category can be realized by endomorphisms, for any given fusion rule, does the method used in 1506.03546 (built on earlier work by people including yourself) essentially eliminate the need to solve the pentagon? Or are there other complications in this approach, such as perhaps it is not clear which algebra to realize the endomorphisms on? And for a specific choice of the algebra (Leavitt in 1506.03546), one might not find all possible fusion categories for a given fusion rule? $\endgroup$
    – Ying
    Commented May 29, 2021 at 16:14
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    $\begingroup$ There are certainly cases where it’s easier to construct the category this way instead of solving pentagons, most notably generalizations of the Haggerup category in Izumi’s work. However, in general yes you run into the problem that the algebra might be quite complicated and so this approach won’t be very helpful. $\endgroup$ Commented May 29, 2021 at 16:33

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