Is 'abelian' necessary in the definition of fusion category? I noticed that there are different definitions of fusion category in different textbook; In 'Tensor Category'by EGNO, a fusion category is necessarily a abelian category; But in some other textbooks , there is no 'abelian' in the definition;
So, I want to known if we do not require fusion category must be abelian, what difference would it make? What will be the impact?
 A: Some authors use $\mathbb K$-linear to simply mean enriched in $\mathsf{Vec}_{\mathbb K}$, see e.g. https://arxiv.org/abs/0803.3652, https://arxiv.org/abs/2003.13812 and all the other articles cited in this answer.
Others use $\mathbb K$-linear to mean enriched in $\mathsf{Vec}_{\mathbb K}$ AND abelien, though this is less common and I can't seem to find an example at the moment.
As indicated by @Jeroen, often the word semisimple requires the category to be abelien.

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*From Müger: "A semisimple category is an abelian category where every short exact sequence splits." (in https://arxiv.org/abs/0804.3587)

*From Bakalov & Kirillov Jr.: "An abelian category $\mathcal C$ is called semisimple if any object $V$ is isomorphic to a direct sum of simple ones $V\cong\bigoplus_{i}V_i$." (in https://www.math.stonybrook.edu/~kirillov/tensor/tensor.html)

Thus the main question is what should it mean for a non-abelian category to be semisimple?  Müger's definition will not work since exact sequences don't make sense, but B&K's definition works as long as we can agree on what a simple object is.  One problem is that in non-abelian categories, there is a difference between simple and cosimple objects: no nontrivial subobjects vs no nontrivial quotient objects.
In categories with an epi-monic factorization (basically when all maps have images), you can recover a weak version of Schur's Lemma: Any nonzero morphism from a cosimple object to a simple object is an isomorphism.  With no such epi-monic factorization, I think that Schur's Lemma breaks down completely.  The B&K style definition of semisimplicity would imply that all cosimple objects must be simple, but there appears to be no reason why the simple objects would need to be cosimple.
The category of tilting modules for $\mathcal U_q(\mathfrak{g})$ is rigid and closed under tensor products, direct sums and summands, so it behaves a bit like a fusion category.  However, if I'm not mistaken it has many more morphisms whose kernels and cokernels are not tilting (i.e. don't exist in the category).  I'm not very familiar with tilting modules so take this with a grain of salt, but these are the sort of categories I would think of as being fusion-like, without being abelian.
