# What is the name of this categorical construction?

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?

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