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If $\mathcal{C}$ is a skeletally small (i.e. it is equivalent to a small category) preadditive category, then we can make the following construction:

First we form the additive category $\text{Mat} \mathcal{C}$ whose objects are $n$-tuples of objects in $\mathcal{C}$ and whose morphisms between these $n$-tuples are appropriated matrices which entries are morphisms in $\mathcal{C}$. In this case $\mathcal{C}$ can be viewed as a full subcategory of $\text{Mat} \mathcal{C}$ in a canonical way, and $\text{Mat} \mathcal{C}$ is also skeletally small.

Next we use Cauchy Completion to obtain an additive idempotent complete category $\widetilde{\text{Mat} \mathcal{C}}$ in the following way: let $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ be the (abelian) category of contravariant additive functors from $\text{Mat} \mathcal{C}$ to the category $\text{Ab}$ of abelian groups. Through the Yoneda Embedding we can identify $\text{Mat} \mathcal{C}$ with the full subcategory of $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ consisting of the representable functors. Then we take $\widetilde{\text{Mat} \mathcal{C}}$ to be full subcategory of $((\text{Mat} \mathcal{C})^{op}, \text{Ab})$ consisting of all retracts of representable functors (in this case these functors will be the direct summands of representable functors). In this case $\widetilde{\text{Mat} \mathcal{C}}$ is the idempotent completion of $\text{Mat} \mathcal{C}$.

Therefore, with this process we obtain an additive idempotent complete category $\widetilde{\text{Mat} \mathcal{C}}$ which canonically contains $\mathcal{C}$ as a full subcategory. Consequently, with this identification we have $\widetilde{\text{Mat} \mathcal{C}} = \text{add} \mathcal{C}$, where $\text{add} \mathcal{C}$ is by definition the full subcategory of $\widetilde{\text{Mat} \mathcal{C}}$ consisting of all direct summands of finite direct sums of objects in $\mathcal{C}$.

Thus this construction give us an additive idempotent complete category $\mathcal{A}$ which contains $\mathcal{C}$ and such that $\text{add} \mathcal{C} = \mathcal{A}$.

Question: does this construction appears in the Mathematical literature? If yes, could you give me some references, and maybe explain what is its use?

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This is the Cauchy completion of $\mathcal{C}$ as an $\mathrm{Ab}$-enriched category.

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