Is there a theory of decomposition into indecomposables? What's the relation to idempotents?

Question

Call a nonzero object of a pointed category simple if it has no proper quotients, and indecomposable if it's not the product of two objects (dual to connected).

Idempotents seem to pop up in many discussions about indecomposability (and I was told they're also relevant to simplicity). I was wondering - is there a nice theory conceptually explaining the relation between idempotents, indecomposability, and simplicity?

Very tentatively I'm hoping for something in the setting of semi-abelian categories because they admit a Jordan-Holder theorem, but this may be a fake and irrelevant justification.

Answer 1

Semi-abelian feels like a red herring here, since you don't even have a direct sum necessarily. What would indecomposable even mean in that context?

In an abelian category, any direct sum decomposition $A\oplus B$ is equipped with a projection map to $A$, and an inclusion of $A$ (using the usual universal properties). The composition of these gives an idempotent endomorphism of $A\oplus B$ from which you can reconstruct the decomposition, and you can easily get a decomposition from an idempotent by taking the image plus the kernel. In particular, in an abelian category, an object is indecomposable if and only if its endomorphism algebra has no idempotents other than 0 or 1.

In a more general additive category, this isn't necessarily true (consider free modules over a ring that has non-free projectives) since the image and kernel of idempotent may not be well behaved. You can fix this by passing to the Karoubian envelope (or idempotent completion) where you formally add an image to every idempotent endomorphism. Generally, indecomposable objects will not be very well behaved outside the Karoubian context.