The finite cartesian powers of $\Delta[1]$ form a cubical object in simplicial sets, inducing a "cubical nerve" functor $N\_\Box: sSet \to Set^{\Box^{op}}$. 


* $N\_\Box$ is a right Quillen equivalence, so it preserves weak equivalences between Kan complexes.
* Cisinski also showed that $N\_\Box$ preserves weak equivalences between nerves of categories.


**Question:** Does $N\_\Box$ preserve arbitrary weak equivalences of simplicial sets?


In other words, does $N\_\Box X$ always have the correct homotopy type? If not, what is an illustrative counterexample? Conversely, is there any larger class of simplicial sets than the union of Kan complexes and nerves of categories (a rather strange class of simplicial sets!) for which $N\_\Box$ computes the correct homotopy type?


Cisinski's work (and, classically, the Quillen equivalence) was in the context of plain cubical sets, but his results (and the Quillen equivalence) extend easily to cubical sets with connections and/or extensions (which permute the different axes of a cube), though different ideas would be needed to get a nerve valued in cubical sets with reversals (which reverse the direction of a cube along a given axis). I'd be interested in what can be said in any of these settings.



Does the cubical nerve preserve weak equivalences of simplicial sets?
Yes. Proposition 8.4.28 of Cisinski's book *Les préfaisceaux comme modèles des types d'homotopie* states that the cubical nerve functor you describe preserves and reflects weak homotopy equivalences. By tracing through Cisinski's proof, one finds that the same is true for cubical sets with connections.


