The groupoid completion of $\mathcal{C}$1 is the pair $\webleft (\mathrm{K}_{0}\webleft (\mathcal{C}\webright ),\iota _{\mathcal{C}}\webright )$ consisting of
- A groupoid $\mathrm{K}_{0}\webleft (\mathcal{C}\webright )$;
- A functor $\iota _{\mathcal{C}}\colon \mathcal{C}\to \mathrm{K}_{0}\webleft (\mathcal{C}\webright )$;
- Given another such pair $\webleft (\mathcal{G},i\webright )$, there exists a unique functor $\mathrm{K}_{0}\webleft (\mathcal{C}\webright )\overset {\exists !}{\to }\mathcal{G}$ making the diagram
commute.
1Further Terminology: Also called the Grothendieck groupoid of $\mathcal{C}$ or the Grothendieck groupoid completion of $\mathcal{C}$.
2See Item 5 of Proposition 9.4.3.1.3 for an explicit construction.
