TITLE: Relation between Ind-completion and "additive"-ind-completion
QUESTION [11 upvotes]: Suppose that $\mathcal{C}$ is a skeletally small additive category.
To enlarge $\mathcal{C}$ and produce a bigger category whose "small" objects can be identified with those in $\mathcal{C}$, one may consider the ind-completion $\operatorname{Ind}\mathcal{C}$ of $\mathcal{C}$ in the sense of Grothendieck and Verdier. Denote by $\operatorname{Fun}(\mathcal{C}^{op},\operatorname{Set})$ the category of all functors from $\mathcal{C}^{op}$ to $\operatorname{Set}$. Then $\operatorname{Ind}\mathcal{C}$ is defined as the full subcategory of $\operatorname{Fun}(\mathcal{C}^{op},\operatorname{Set})$ whose objects are the filtered colimits (in $\operatorname{Fun}(\mathcal{C}^{op},\operatorname{Set})$) of representable functors.
We may also consider the category $\operatorname{AddFun}(\mathcal{C}^{op},\operatorname{Ab})$ of additive functors from $\mathcal{C}^{op}$ to the category of abelian groups and define $\operatorname{AddInd}\mathcal{C}$ as the full subcategory of $\operatorname{AddFun}(\mathcal{C}^{op},\operatorname{Ab})$ whose objects are the filtered colimits (in $\operatorname{AddFun}(\mathcal{C}^{op},\operatorname{Ab})$) of representable functors. This construction is thoroughly studied, for instance, in the paper "Locally finitely presented additive categories", by W. Crawley-Boevey.
I am trying to understand the relation between $\operatorname{Ind}\mathcal{C}$ and $\operatorname{AddInd}\mathcal{C}$. I have seen in two papers statements claiming that $\operatorname{Ind}\mathcal{C}$ and $\operatorname{AddInd}\mathcal{C}$ are equivalent, but do not yet understand why this should be true. Any references or hints would be very welcome.

REPLY [6 votes]: The equivalence you mention holds more generally whenever your base of enrichment has a finitely presentable unit. This certainly includes $Ab$ but also many other examples: $Cat$, $sSet$, $GAb$, $DGAb$, etc.
Assume that $\mathcal V=(\mathcal V_0,\otimes,I)$ is a symmetric monoidal closed complete and cocomplete category. Then you can define the Ind-completion of a $\mathcal V$-category $\mathcal  C$ as the closure of the representables in $[\mathcal C^{op},\mathcal V]$ under (conical) filtered colimits; let's call it simply $Ind(\mathcal C)$, and denote by $J:\mathcal C\to Ind(\mathcal C)$ the inclusion.
This free cocompletion can be recognised as follows (see e.g. 4.2 from "Notes on enriched categories with colimits of some class" by Kelly and Schmitt): A $\mathcal V$-functor $F:\mathcal C\to\mathcal K$ exhibits $\mathcal K$ as the Ind-completion of $\mathcal C$ if and only if:

$F$ is fully faithful;
$\mathcal K$ has filtered colimits;
$Fc$ is finitely presentable in $\mathcal K$ for any $c\in\mathcal C$ (i.e. $\mathcal K(Fc,-):\mathcal K\to \mathcal V$ preserves filtered colimits);
The closure of $\mathcal C$ in $\mathcal K$ under filtered colimits is $\mathcal  K$ itself.

Now assume that the unit $I$ of $\mathcal V$ is finitely presentable, that is $\mathcal V_0(I,-):\mathcal V_0\to Set$ preserves filtered colimits. Let $\mathcal C$ be a $\mathcal V$-category and $J:\mathcal C\to Ind(\mathcal C)$ be the inclusion in its enriched Ind-completion (then $J$ satisfies (1)-(4) above). We can then consider the underlying ordinary functor $J_0:\mathcal C_0\to Ind(\mathcal C)_0$ of $J$; then the properties (1), (2), and (4) still hold for $J_0$ (independently of the unit) and (3) holds since
$$ Ind(\mathcal C)_0(J_0c,-)\cong \mathcal V_0(I,Ind(\mathcal C)(Jc,-)_0) $$
is a composite of filtered-colimit-preserving functors.
It follows by the characterization above of Ind-completions (for $\mathcal V=Set$) that $Ind(\mathcal C)_0$ is the Ind-completion of $\mathcal C_0$; in other words
$$ Ind(\mathcal C)_0\simeq Ind(\mathcal C_0).$$