# The nerve of categories preserves weak equivalence?

Does the nerve functor $\operatorname{Cat}\to \operatorname{sSet}$ from small categories to simplicial sets preserve weak equivalences?

If $f\colon C\to D$ and $g\colon D\to C$ are functors of small categories such that $\alpha\colon f\circ g\sim 1_D$ and $\beta\colon g\circ f \sim 1_C$, then we have a pair of maps between the corresponding simplicial sets. What can we say about $\alpha$ and $\beta$ under the nerve functor? (Homotopic in some sense?)

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If one uses the definition of natural transformation in terms of the interval category then one can see that adjoint functors between categories give rise to a homotopy equivalence between the nerves and so in particular an equivalence of categories gives a homotopy equivalence. – Callan McGill Jul 2 '13 at 13:31
@CallanMcGill Thanks a lot. My reasoning is the following: If $\tau: f\Rightarrow g: C\to D$ is a natural transformation. Then the nerve induces $F\Rightarrow G$ a left homotopy? – Ma Ming Jul 2 '13 at 13:38

The proposition in question says: assume $C, D$ are topological categories and $F,G$ continuous functors from $C$ to $D$. Now, any natural transformation $\eta:F \implies G$, induces a homotopy between the continuous functions $BF, BG$ on classifying spaces $BC \to BD$.
In order to use this proposition, treat $C$ and $D$ as topological categories with the discrete topology on each hom-set, so all functors are continuous.