Simple question about idempotent completion

Question

I am learning about the idempotent completion (Cauchy completion) of a category.

I wonder, is the category of finite sets the idempotent completion of something?

Thanks!

Answer 1

An idempotent complete category $\mathcal{C}$ is the idempotent completion of a category $\mathcal{D}$ iff $\mathcal{D}$ is equivalent to a full subcategory $\mathcal{C}_0\subseteq\mathcal{C}$ such that every object of $\mathcal{C}$ is a retract of an object of $\mathcal{C}_0$. When $\mathcal{C}=\mathtt{FinSet}$, this just means that $\mathcal{C}_0$ must contain the empty set and arbitrarily large finite sets (since a finite set $F$ is a retract of a finite set $G$ iff either $F$ and $G$ are both empty or $F$ and $G$ are both nonempty with $|F|\leq|G|$). So up to equivalence, the categories that $\mathtt{FinSet}$ is the idempotent completion of are the categories of all finite sets whose cardinalities are in some set $A\subseteq\mathbb{N}$, where $A$ is infinite and $0\in A$.