I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple objects $X_i$ in $\mathcal{C}$ are those that have $\mathcal{C}(X_i, X_j) \cong \delta_{i,j} k$, where $\mathcal{C}$ is $k$-linear (that does not imply abelian, though)?
Then, some light at the issue: On the first page, it says that being simple in an abelian and $k$-linear category implies that $\mathcal{C}(X_i, X_j) \cong \delta_{i,j} k$.
But then, in a Kuperberg article, he calls an object $X$ "strongly simple" if $\mathcal{C}(X,X) \cong k$, somehow implicitly implying that this property implies simplicity, and not the other way around.
What's the right definition now? In case that abelian categories and linear categories are too different, I care for the case needed to define fusion categories (and only for the fields $\mathbb{R}$ and $\mathbb{C}$).
The two definitions are not equivalent, and the first does not imply the second; in a $k$-linear abelian category, any division algebra $D$ over $k$ can occur as the endomorphism $k$-algebra of a simple object $X$ (e.g. in the category of right $D$-modules).
If $k$ is required to be algebraically closed (which I think is a standard simplifying assumption when dealing with fusion categories), then the only finite-dimensional endomorphism algebra that can occur is $k$ itself, and if $X$ is in turn required to be dualizable (which is part of the definition of a fusion category) then I believe this implies that $\text{End}(X)$ is finite-dimensional.