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 builing 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?

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?

That's the second part of my venture to become more comfortable with the concept of idempotent elements and idempotent splittings from category theoretical viewpoint. In 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 $R−ModFree$ of free $R$-modules taking it's 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 occure naturaly as builing 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 start with certain category $C$ (construct from this one another say $F(C)$ and then pass to it's idempotent completion $\widehat{F(C)}$.

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

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

Of course, that's just an exacmple but similar strategies occure 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 neccessary 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 $R-Mod$ example in mind was the "neccessarity of projective objects in order to stude right exact functors.

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

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 builing 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?