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Qiaochu Yuan
Qiaochu Yuan
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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 a canonicalthe 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.

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, biproducts are an absolute colimit.

In general we can ask for a canonical 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.

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.

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Qiaochu Yuan
Qiaochu Yuan
  • 118.2k
  • 40
  • 447
  • 741

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, biproducts are an absolute colimit.

In general we can ask for a canonical 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.