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I was dissatisfied with the discussion at semisimple category because it only defined a semisimple monoidal Vect-enriched category, completely ignoring the more common notion of semsimple abelian category.
So, I stuck in the definition of semisimple abelian category.
However, I still think there is a lot that could be improved here: when is a semisimple abelian category which is also monoidal a semsimple monoidal category in some sense like that espoused here???
I think this article is currently a bit under the sway of Bruce Bartlett’s desire to avoid abelian categories. This could be good in some contexts, but not necessarily in all!
We don’t hear a lot from Bruce here, so I’m wondering who was under his sway… :-) Why does Bruce desire to avoid abelian categories?
The article was under Bruce’s sway. He explains there why he wants to avoid abelian categories, but I sympathize too little to try to convey his point. I guess he’s trying to say that for semisimple categories, all that really matters are biproducts and splitting idempotents. Sounds vaguely familiar, eh? But I’m still not sure I agree…
The second definition of semisimple category on this page is as a “monoidal linear category” with certain properties. But none of the properties or any of the later discussion on this page uses the monoidal structure, nor is it very consistent with the first defintion (as a semisimple abelian category, which makes no reference to monoidality).
There’s a kind of moral conflict built into our treatment of semisimple category and semisimple object, and I’m not yet sure what’s the standard way to fix it. In semisimple object we say
An object X in an abelian category A is said to be semisimple or completely reducible if it is a coproduct (direct sum) of simple objects.
In semisimple category we say
A semisimple category is a category in which each object is a direct sum of finitely many simple objects, and all such direct sums exist.
The issue is that the first one has no finiteness condition while the second one does. So, with these definitions we can have an abelian category that is not semisimple, where every object in it is semisimple. For example: the category of vector spaces over any field.
The moral conflict becomes an actual contradiction when in semisimple category we write
Definition 2.1. (semisimple abelian category) An abelian category is called semisimple if every object is a semisimple object, hence a direct sum of finitely many simple objects.
If we omit the finiteness condition in the definition of semisimple category or add it to the definition of semisimple object, then a semisimple category becomes the same as a category where every object is semisimple.
The definition of “semisimple object” is consistent with the definition of “simple module” in Anderson and Fuller’s Rings and Categories of Modules: that is, they say a semisimple module is a possibly infinite coproduct of simple modules. So, I don’t think we should change that. Is it standard to require a finiteness condition in the definition of semisimple category?
I see that in this meditation on various definitions of “semisimple category”, none of the definitions impose the finiteness condition I am worrying about:
So, I am tempted to remove it.
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