Let $(\mathcal{C}, \otimes, \oplus, I, a, r, l)$ be an abelian semisimple monoidal category, then is it necessary that all module categories over $\mathcal{C}$ be semisimple?

I can convince myself that they need to be additive and have indecomposable objects, but I cannot see semisimplicity.


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    $\begingroup$ I don't really work with module categories over monoidal categories, but shouldn't you have some 'finite-dimensionality' condition on your module category for something like this to hold? Also, you do assume that $\mathcal{C}$ is semisimple right? (it's written in the title but not in the body of your post). $\endgroup$ – Mathematician 42 Feb 13 '18 at 20:37
  • $\begingroup$ I edited my question, thank. Finite dimensionality for hom-set you mean? $\endgroup$ – Fatimah Feb 13 '18 at 21:06
  • $\begingroup$ Can you give me a reference where you studied these things? I'm not familiar enough with these concepts to answer straightaway. If no-one else answers, I might think about it. $\endgroup$ – Mathematician 42 Feb 13 '18 at 21:15
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    $\begingroup$ Well, taking $\mathcal C:=\mathbf{finVect}_k$, wouldn't that mean that every (finitely generated, associative, unital) algebra over $k$ is semisimple? $\endgroup$ – Berci Feb 14 '18 at 0:02
  • $\begingroup$ @Mathematician42 Thanks! Basically this one: arxiv.org/abs/math/0111139v1 $\endgroup$ – Fatimah Feb 22 '18 at 12:13

No, they usually are not. To elaborate on Berci's comment, if $k$ is a field and $\mathcal{C}$ is the usual monoidal category of (finite-dimensional) vector spaces over $k$, then for any $k$-algebra $A$, the category of (finitely generated) right $A$-modules is a module category over $\mathcal{C}$ (by just using the usual tensor product over $k$: if $V$ is a vector space and $M$ is an $A$-module, then $V\otimes_k M$ is again an $A$-module). This category is semisimple iff $A$ is a semisimple ring, so any non-semisimple $k$-algebra gives a counterexample (e.g., $A=k[x]$, or $A=k[x]/(x^2)$ if you want some stronger finiteness conditions).

(It's not clear to me what kind of finiteness assumptions you want to be making here. If your definition of "semisimple" requires every object to be a finite coproduct of simple objects, for instance, then you can get an even easier example by just taking the category of all vector spaces as a module category over the category of finite-dimensional vector spaces. In that case, it's completely unreasonable to hope for a module category to be semisimple unless you also impose some finiteness assumptions on it.)


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