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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
up vote 6 down vote accepted

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.

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.

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