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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?

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  • $\begingroup$ The point of the idempotent completion is so that direct summands of objects of your category are now objects as well. $\endgroup$ – Sam Hopkins Feb 28 at 18:42
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    $\begingroup$ Idempotents appear very naturally when studying the cohomology of algebraic varieties. For example if some finite group acts on your variety then you can consider the piece of cohomology cut out by characters of this group (corresponding to idempotents in the group algebra). $\endgroup$ – François Brunault Feb 28 at 18:47
  • $\begingroup$ @Sam Hopkins: that's true. The point is when we think about the couple of constructions I mentioned (e.g. the motivic case) where it become neccessary to deal with a category beeing closed under taking direct summands? What might fail if not not do it? $\endgroup$ – MortyPB Feb 28 at 18:49
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    $\begingroup$ I think a very illuminating example is the construction of Deligne's category $\mathrm{Rep}(S_t)$ of "representations of the symmetric group $S_t$" where $t$ is a complex parameter. The idea is that for integer $n$, every representation of $S_n$ is a direct summand of a tensor power of the defining representation. So to mimic this for an arbitrary parameter $t$, first you create via diagrammatic rules a category whose objects correspond to tensor powers of the defining representation; then you take the Karoubian envelope to get all representations. $\endgroup$ – Sam Hopkins Feb 28 at 18:56
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    $\begingroup$ Another example for motives: you need projectors in order to define even very basic objects. For example the Tate motive $\mathbb{Z}(1)$ (or rather its inverse, the Lefschetz motive), you need to decompose the cohomology of the projective line (or the multiplicative group). If you don't have the Tate motive then you can't define motivic cohomology for example. $\endgroup$ – François Brunault Feb 28 at 19:30
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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).

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  • $\begingroup$ About the "deep yoga": According to the nLab, the karoubi envelope is a special case of the cauchy completion of a category. This completion has some characterisations that are not related specifically to idempotents. ncatlab.org/nlab/show/… $\endgroup$ – Gerrit Begher Mar 8 at 2:32
  • $\begingroup$ I have extended my answer. I suspect that the "yoga things" mentioned in the comments is not really actual for motives. $\endgroup$ – Mikhail Bondarko Mar 8 at 6:35
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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.

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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.)


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

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  • $\begingroup$ Yes, Jannsen result is not very old; that is why I did not put it on the first place in my answer.:) A terminological remark: I think that it is better to speak about numerical motives than about Chow motives modulo numerical equivalence. Moreover, Chow motives are certainly useful, and one does not realy want to replace them by an abelian category. $\endgroup$ – Mikhail Bondarko Mar 9 at 12:20

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