Computing the Ind-completion of the terminal category

Question

Let $C$ be any category. $\text{Ind}(C)$ is the free completion under all filtered colimits.

Can you help me in computing the Ind-completion of the terminal category?

Answer 1

The Ind-completion may be constructed as the closure of the representables in $\widehat{C}$ under filtered colimits. But the subcategory of representables in $\widehat{*}$ is a terminal category, so every functor into it is constant. Any filtered category $J$ is connected, so the colimit of a constant $J$-indexed functor at the representable $y*$ is just $y*$ itself. That is, the terminal category is its own Ind-completion.

To confirm, we can check the universal property. Given any functor $F:*\to D$, where $D$ has filtered colimits, the claim is that there exists a unique functor $*\to D$ extending $F$ and preserving filtered colimits; in other words the claim is simply that $F$ automatically preserves filtered colimits. But this follows by the same argument as above: namely, that the filtered colimit of a constant diagram on some object $x$ is simply $x$.