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.

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    $\begingroup$ Of course, there is a natural collection of simplicial sets containing Kan complexes and nerves of categories ;) $\endgroup$ Feb 18 '19 at 5:13
  • $\begingroup$ Is there a reference for cubical sets with extensions? $\endgroup$ Feb 18 '19 at 8:38
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    $\begingroup$ The main tool here is Prop. 4.3.12 in Astérisque 308. This proves that the "cubical nerve" as above, but also its version with connections do preserve and detect weak equivalences. It also implies right away that there is a symmetric version (from symmetric simplicial sets to cubical sets with reversals), and that forgetting connections also preserve and detect weak equivalences. Since the link between simplicial sets and symmetric simplicial sets is also documented (prop. 8.3.8 in loc. cit.), this says that all versions you could naturally come with will be homotopically well behaved. $\endgroup$ Feb 18 '19 at 8:48
  • $\begingroup$ It seems I didn't read carefully enough, because I had the impression that 4.3.12 required the nerve to be induced by a functor between the sites (which this one is not). But no, it is more powerful than I realized -- thanks! $\endgroup$
    – Tim Campion
    Feb 18 '19 at 14:48
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    $\begingroup$ It seems I was overconfident: Prop. 4.3.12 does not imply that forgetting connections preserves weak equivalences (all other assertions are correct though). It is true that forgetting connections preserves weak equivalences, but we have to adapt the argument (a monoidal (as opposed to Cartesian) version of 4.3.12). $\endgroup$ Feb 18 '19 at 19:51

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.

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    $\begingroup$ Wow, I thought I'd gone over this section carefully, but apparently I missed this too -- thanks! $\endgroup$
    – Tim Campion
    Feb 18 '19 at 14:50

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