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?)



The nerve of categories preserves weak equivalence?
Such questions are typically framed in terms of Classifying Spaces, but the answer is yes. It follows from, for instance, from Proposition 2.1 in Graeme Segal's article 
*Classifying spaces and spectral sequences*. Publications Mathématiques de l'IHÉS, 34 (1968), p. 105-112, available [here](http://www.numdam.org/item?id=PMIHES_1968__34__105_0).


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. 


