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Chris Schommer-Pries posted a question/suggestion in the query box at semisimple category
I have touched the formatting at semisimple category slightly, and hereby I am moving that old query box from there to here:
begin forwarded discussion
What does it mean to ’have subobjects’? (I assume that the ’direct sums’ are biproducts.) —Toby
have subobjects = idempotents split and yes, finite biproducts. Simple objects are ones in which End(X) = k.
Urs: shouldn’t we say something like: a category is semisimple if each object is a direct sum of finitely many simple objects?
Bruce: Urs, you’re right, and that’s indeed the way one morally thinks about it, but it’s a less canonical way of proceeding. We ask ourselves: given a linear category with direct sums and subobjects, and a chosen maximal collection $\{X_i\}$ of nonisomorphic simple objects, how can we check if its semisimple? In the one way, we have to check whether a certain canonically defined map is an isomorphism. In the other way, we have to check if each object $V$ can be expressed as a direct sum of the $X_i$’s. Actually finding such a decomposition would be a noncanonical operation. So your shorter more snappy definition would force an auditor to perform an evil thing if he actually wanted to check it :-) It’s that old thing about “only that part of a representation which behaves like an irreducible $\rho$ is canonical, the actual break-down of that rep into direct sums of $\rho$’s is noncanonical”. That is, what is canonical is $Hom(V, X_i)$ and not $V = \bigoplus_i n_i X_i$.
Also, if we just had “a category is semisimple if each object is a direct sum of finitely many simple objects” without the conditions on direct sums and subobjects then we could have someone who nastily removes, say, all three-dimensional vector spaces from $Vect$. It would still satisfy “each object is a direct sum of finitely many simple objects” but it shouldnt be regarded as a semisimple category since there are “holes”.
Urs: okay, so “each object is finite sum of simples” is the right idea, while the right defintion is a bit different. I have accoridngly now created a section “Idea” with the former statement. (Every entry should start with a section “Idea”!) See if you like this. Otherwise, feel free to adjust.
But in any case, it would be nice to have a discussion on how the “right definition” implies that every object is isomorphic to a finite direct sum of representables.
Chris Schommer-Pries: Shouldn’t you require that for simples, End(X,X) is a simple algebra, not necessarily the ground field? For example the category of H-modules where H is the quaternion algebra over the reals. Shouldn’t this be semi-simple?
Chris: It is not standard to assume the direct sum decomposition is finite in defining a semisimple category. For instance, the category of all vector spaces is semisimple. What’s written here at present looks like a definition of semisimple for categories where it has already been assumed that every object has finite length.
end forwarded discussion
I agree with Chris at the end here. We should not restrict ourselves to finite sums, although that might be an interesting variation. ($\kappa$-semisimple, anybody?)
There are different traditions. In references like those listed at FFRS-formalism “semisimple” implies the finiteness clause.
(Though just checking with their latest I see that they did switch to saying “finitely semismple” now.)
they did switch to saying “finitely semismple”
That seems like a reasonable use of language. Of course, in some contexts, that could be the default, just like ‘ring’ means something commutative to some people.
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