Magic behind idempotent-complete categories a.k.a. why (sometimes) be Karoubian is sexier than be Abelian

Question

It is well know that Karoubian categories (also called idempotent-complete categories) are living between additive and Abelian categories. While one of the most famous advantages to work with abelian cetegories is that they are closed under building kernels and cokernels of arbitrary morphism, the Karoubian categories have a slightly weaker property that only idempotent morphisms (i.e. $p$ with $p^2=p$) from there share this property.

Nevertheless it seems that in diverse constructions Karoubian categories or Karoubian envelopes of additive categories provide a more natural setting than Abelian categories.

I hope my question not becomes too broad: What is the philosophical meaning behind Karoubian categories or say in simpler words why in some constructions they are more prefered (e.g. category of pure motives and in K-theory) in contrast to say at first glance more 'flexible' abelian categories?

My natural guess is that if we think about the construction of of the category of pure motives we start with the category of smooth varieties over a base field and pass after application of this magic Karoubian completion functor to
idempotent-complete categories. Since it's not abelian it contains by definition only kernel and cokernels of idempotent morphisms but that's all we need there to proceed the constrution.

This lead me to conjecture that the main advantage of Karoubian categories in contrast to Abelian categories mights show when one have to perform a construction where one starts with a certain preadditive category but the construction requireres a category having at least some kernels and cokernels.

Now one can pass canonically to the Karoubian completion or extend the initial category to an Abelian category. But exactly here I see an obstacle with the secound and at first glance more 'natural' approach:

Does there always a way to embedd a preadditive category in an abelian category? If yes, seemingly the disadvantage of this approach seems to be that this Abelian category is much harder to control, while the Karoubian completion is constructed quite canonically and behaves more 'similar' to initial preadditive category.

Questions: Is what I tried to sketch above exactly the motivation why Karoubian categories are in some constructions more prefered then Abelian categories?

Are there more reasons making Karoubian categories 'interesting'? Is there any intuition or important example one should have in mind how to think about Karoubian categories?

Answer 1

I don't know if this is going to answer your question but here's some relevant background. Splitting idempotents has a very special property from a categorical point of view: it is an absolute colimit (and also an absolute limit), meaning that it is preserved by any functor whatsoever. Two somewhat more familiar examples of absolute colimits, in enriched settings:

• In categories enriched over pointed sets (equivalently, categories with zero morphisms), zero objects are an absolute colimit. This also applies to $\text{Ab}$-enriched categories.
• In linear ($\text{Ab}$-enriched / preadditive) categories, (finite) biproducts are an absolute colimit.

In general we can ask for the completion of a (possibly enriched) category under absolute colimits; this is called its Cauchy completion because it turns out to specialize to Cauchy completion when thinking of metric spaces as enriched categories. We have:

• For ordinary ($\text{Set}$-enriched) categories, the Cauchy completion of $C$ is given by splitting all idempotents, or equivalently by taking the full subcategory of the presheaf category $\widehat{C} = [C^{op}, \text{Set}]$ on the tiny objects (objects such that $\text{Hom}(F, -)$ preserves all colimits; these turn out to be exactly the retracts of representables).
• For linear ($\text{Ab}$-enriched) categories, the Cauchy completion of $C$ is given by taking all formal direct sums and then splitting all idempotents, or equivalently by taking the full subcategory of the presheaf category $\widehat{C} = [C^{op}, \text{Ab}]$ on the tiny objects (these turn out to be exactly the retracts of finite direct sums of representables).

In particular, if we consider a ring $R$ as a one-object linear category $BR$, its Cauchy completion is exactly the category of finitely generated projective $R$-modules, which we famously use to define K-theory. This construction is a complete Morita invariant in the following sense: two rings $R, S$ are Morita equivalent (meaning $\text{Mod}(R) \cong \text{Mod}(S)$) iff their categories of finitely generated projective modules are equivalent (and this generalizes to enriched categories and Cauchy completions).

Cauchy completion is in some sense the "most harmless" and "most inevitable" completion: if you are ever going to apply a functor from your category $C$ to a category $D$ with colimits then every absolute colimit of objects in $C$ will appear in $D$ anyway (all idempotents will be split, etc.) so you might as well add them in first. Unlike the Yoneda embedding, which is the free cocompletion, Cauchy completion does not destroy colimits that may already exist in $C$. And because absolute colimits are preserved by all Hom functors $\text{Hom}(c, -)$, unlike adjoining colimits in general, the morphisms both into and out of an absolute colimit are already uniquely determined, so you have no choice how to do it anyway.

On the other hand attempting to write down some sort of completion of a linear category producing an abelian category seems quite tricky and potentially unwieldy. We can take the free cocompletion ($\text{Ab}$-valued presheaves) but again this destroys most existing colimits. Maybe the Isbell envelope has better properties but I don't know anything about it. Meanwhile the Cauchy completion is relatively easy to work with.

Answer 2

I'd like to comment that idempotent completion is sometimes the right thing to do 'by design'. (I don't have enough rep to just leave a comment though.) I'm reminded of the category of motives, in which you would like to have some kind of 'cellular decomposition' of your varieties, the archetypical example being that one would like the motive $[\mathbb{P}^1]$ to split as $[\mathbb{A}^1] + [\text{pt}]$ or something along those lines. Idempotent completion does exactly that: it forces the idempotent map $\mathbb{P}^1 \to \operatorname{Spec} k \to \mathbb{P}^1$ to have a kernel, which leads to the desired decomposition.