Quillen Q-construction

Remarks

The pullback in clause 3 exists because admissible monomorphisms are stable under pullback along admissible epimorphisms, and the two composites $N' \times_N P' \rightarrowtail P' \rightarrowtail P$ and $N' \times_N P' \twoheadrightarrow N' \twoheadrightarrow M$ are again admissible. Each admissible monomorphism $i \colon M \rightarrowtail N$ gives the morphism $M \twoheadleftarrow M \rightarrowtail N$ (identity on the left), and each admissible epimorphism $p \colon N \twoheadrightarrow M$ gives the morphism $M \twoheadleftarrow N \rightarrowtail N$ (identity on the right). Quillen proved that $\pi_1(BQ\mathcal{C}, 0)$ is the Grothendieck group $K_0(\mathcal{C})$, so the definition is right in degree $0$.

Example: for a ring $R$, the category $\mathcal{P}(R)$ of finitely generated projective left $R$-modules, with the split short exact sequences, is exact, and $K_n(\mathcal{P}(R))$ is the Quillen K-group $K_n(R)$.

Non-example: in $\mathcal{P}(\Z)$, the diagram $0 \twoheadleftarrow \Z \xrightarrow{\,2\,} \Z$ is not a morphism of $Q\mathcal{P}(\Z)$. Multiplication by $2$ is a monomorphism, but it is not admissible, because its cokernel $\Z/2$ is not a projective $\Z$-module.