Quillen Q-construction

Summary

The Q-construction turns an exact category $\mathcal{C}$ into a new category $Q\mathcal{C}$ with the same objects. A morphism of $Q\mathcal{C}$ is a way to see one object as a subquotient of another. Quillen defined the K-groups of $\mathcal{C}$ as the homotopy groups of the loop space of the classifying space $BQ\mathcal{C}$. This definition works for any exact category, not only for rings, and it gives the main structural theorems of algebraic K-theory: resolution, dévissage and localization.