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briefly added something to fusion category. See also this blog comment.
added pointer to:
added pointer to:
(maybe we should have a dedicated entry for braided fusion categories – not sure)
oh, sorry, this reference should be included via anyonic topological order via braided fusion categories – references – will fix
added references on fusion 2-categories:
Christopher L. Douglas, David J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds [arXiv:1812.11933]
Thibault D. Décoppet, Matthew Yu, Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories [arXiv:2306.08117]
It is known that one can define a fusion category up to equivalence by considering $G$-graded vector spaces for $G$ a finite group and taking the associator to be defined by a cohomology class of $H^3(G,k^{\times})$. I am somewhat confused by the actual construction leading to this statement. Is this to be understood as a construction that takes as input $G$ and a cocycle so that if the cocycles are in the same class then the corresponding fusion categories are equivalent but not isomorphic? Usually the notion of isomorphism of categories is too strong, so it makes me wonder if there is a refined notion of fusion category constructed in a similar way such that they are only equivalent to other fusion categories constructed from the same cocycle, not only the same class.
For reference, I have slightly expanded our statement of this example (now here), which originates probably with Etingof, Nikshych & Ostrik 2005, item 1. on p. 584 (p. 4 in the pdf).
Yes, up to equivalence, these fusion categories should depend on the cohomology class of the 3-cocycle only.
It’s not clear to me how one would reasonably restrict the notion of equivalence such as to retain the actual cocycle. And why would you want to do this?
This comes as a confusion I have regarding some computations on a project related to generalized symmetries I’m currently working on, in particular related to the concept of dual or quantum symmetry. But I can’t really phrase out a concrete question to ask here just yet without first sorting out the issue (that I highlighted couple of times in the nLab entry) that all that literature effectively ignores the need for concretely defining “action” for these things (in a sense, it’s like talking about $G$-symmetries by just concentrating on properties of $G$ and not of a $G$-action). I guess I just got ahead of myself by asking the question in #6.
Is there an analogous concept but for categories enriched over virtual vector spaces? I think this and an analog of Turaev-Viro would turn out to be relevant for e.g. global symmetries of theories quantized in K-theory.
Not that I know of.
But depending on what you have in mind it may be that you could do with super-vector spaces instead of virtual vector spaces?
In that case there is a notion of “superfusion category” and there is a good general theory such as Deligne’s theorem on tensor categories.
Maybe. I was thinking what a functorial field theory valued in virtual vector spaces is supposed to be and whether this would be related to the usual assumptions of positive level on CS/WZW.
Re#7: I got a better answer than #8, goes back to the paper by Majid about octonions as twisted group algebras. One of the main observations there is that the associator of the octonions, a group 3-cocycle $\omega$, is not exactly one (obviously, since they are not associative) but it is a coboundary, so that it defines a trivial cohomology class. If we take the construction of $\text{Vec}_{[\omega]}(\mathbb{Z}_2^3)$, this will not care about $\omega$, only about $[\omega]=1$ and will return something equivalent to just $\text{Vec}(\mathbb{Z}_2^3)$. Since these categories are representation categories of (twisted) group algebras, the statement probably is something like, the untwisted algebra $\mathbb{R}[\mathbb{Z}_2^3]$ and the twisted algebra $\mathbb{R}_{\alpha}[\mathbb{Z}_2^3]$ for $\omega=d\alpha$ are not isomorphic but Morita equivalent? But in any case, the fusion category construction seems to be too coarse to detect these distinctions?
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