Motivation for Karoubi envelope/ idempotent completion

Question

This is the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In the first part we considered the interpretation of idempotent elements & splitting from viewpoint of commutative algebra. The most fruitful analogy (at least for me) was that if we consider the category $\text{$R$-ModFree}$ of free $R$-modules, taking its completion means making it closed under taking direct summands. As the direct summands of free modules are exactly the projective modules completing means "to add some objects" which occur naturally as building blocks.

Now in case of commutative algebra projective objects allows to deal with projective resolutions and provide a framework for direct calculations of derived functors of right exact functors.

I read that there are a lot of constructions spreaded in a relatively wide areas of mathematics where one starts with a certain category $C$, construct from this one another say $F(C)$, and then pass to its idempotent completion $\widehat{F(C)}$.

Probably the most prominent example is the construction of pure motives where we start with category $(\operatorname{Sm}/k)$ of smooth varieties over a field $k$, then pass to category of correspondences $\operatorname{Cor}_k$, build its idempotent completion $\widehat{(\operatorname{Cor}_k)} $ and go ahead with the construction to build the category of Motives $\operatorname{Mot}_k$ and then, by trying to mimic the procedure of building the derived category, we arrive at the category of pure motives (of course that's just a very coarse overview).

The point of my interest is the necessity of taking idempotent completion in the intermediate step.

Of course, that's just an example, but similar strategies occur for example in $K$-theory when one study vector bundles or in constructions dealing with triangulated categories.

My Question: Can there be extracted a common motivation in these examples making the step that takes idempotent completion necessary or does it in every construction almost everywhere strongly depend on "what one wants"?

The only one "general mantra" that I found up to now having the $\text{$R$-Mod}$ example in mind was the necessity of projective objects in order to study right exact functors.

Question: Is this the only motivation or are there some other common deep reasons for the importance of taking idempotent completions?

Answer 1

The "motivic motivation" is that by idempotent completing correspondences over a finite field one obtains a category of homological motives where Kunneth decompositions of diagonals are available. Moreover, over any field the category of numerical motives is abelian semi-simple.

The proof of the latter statement is relatively simple, and can probaly be generalized to other relevant settings. Yet I do not think that there exists any "deep" and general yoga that says that idempotent completions are crucially important (and that is really relevant for motives).

Another observation is that over a field of positive characteristic $p$ we don't know whether Voevodsky motives of arbitrary varieties belong to the (smallest strict) triangulated subcategory generated by motives of smooth projectives, but they belong to the subcategory generated by Chow motives (if the characteristic $p$ is invertible in the coefficient ring).

Answer 2

One very general categorical observation is that the idempotent completion functor can be factorised by first passing from the given linear category $\mathcal{A}$ to the non-unital ring $\bigoplus_{X,Y \in \mathcal{A}}\mathcal{A}(X,Y) $, and then taking the linear category $\mathrm{Idem}(\bigoplus \mathcal{A})$ of idempotent elements. The functors $\bigoplus \dashv \mathrm{Idem}$ form an adjoint pair, and idempotent-complete linear categories are algebras for the resulting monad.

More generally, for a category $\mathcal{C}$ enriched in pointed sets, you first pass to the semigroup $\bigvee \mathcal{C}:= \coprod_{X,Y \in \mathcal{C}}\mathcal{C}(X,Y)$ in pointed sets, then take the category of idempotent elements.

On some level, this means that idempotent completion is the universal way to obtain invariants which are really non-unital in nature. You can recover things like idempotent-complete module categories of a unital ring $R$ without knowing its unit.

Answer 3

My understanding of the use of Karoubian completion for motives is that one would really like to have an abelian category of pure motives (modulo homological equivalence, say). However, we don't know how to adjoin all kernels and cokernels, and the Karoubian completion is the best we can do.

There is a hope for an abelian category of pure motives that has all the nice properties we want. There are many flavours of motives around (Chow, André, Nori, Voevodsky, ...), and each of them satisfies some but not all of the desired properties. You use whichever one is most convenient for your problem.

(As Mikhail Bondarko pointed out: Chow motives modulo numerical equivalence¹ are semisimple abelian, and this is basically the only way we know how to prove Chow motives form an abelian category. However, this result of Jannsen was only proven in 1992, so I don't think it was the original motivation.)

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¹The problem with Chow motives modulo numerical equivalence is that it does not have a cohomological realisation, unless we prove standard conjecture D.