Kerodon

Example 7.2.3.3 (Quillen's Theorem A). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Suppose that, for every object $X \in \operatorname{\mathcal{D}}$, the category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ has weakly contractible nerve. Applying Theorem 7.2.3.1, we deduce that the induced morphism of simplicial sets $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is right cofinal. In particular, it is a weak homotopy equivalence (Proposition 7.2.1.5). This recovers a classical result of Quillen (see [MR2655184]).