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In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified simplicial sets spanned by the strict complicial sets is equivalent to the category of globular strict $\omega$-categories.

A troubling corollary of these two facts together is that upon passing to the homotopy category, since all strict complicial sets are both fibrant and cofibrant, that all pseudofunctors between strict $\omega$-categories are representable by strict $\omega$-functors.

Is there a general coherence result for strict $n$-categories showing that indeed every pseudofunctor can be strictified up to a sufficiently weak notion of pseudonatural homotopy? (I am defining a pseudofunctor $A\to B$ of strict $n$-categories following Garner to be a strict $n$-functor $C(A)\to B$ where $C$ is the cofibrant replacement comonad arising from the Free-Forgetful adjunction to $n$-computads).

If not, has anyone conjectured a class of monomorphisms of stratified simplicial sets at which we could localize in order to obtain the correct model category?

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    $\begingroup$ I am cautiously assuming that there is no such coherence result, and if that is the case, what is missing is that not all equivalences of higher arrows are marked. The answer could be localizing at the inclusions $\Sigma^n(J)\hookrightarrow \Sigma^n(\bar{J})$ where $J$ is the stratified Street nerve of the contractible groupoid with two objects, $\bar{J}$ is the entire superset where all $1$-cells are marked, and $\Sigma$ is the two-point suspension. Just a guess though. $\endgroup$
    – Harry Gindi
    Commented May 3, 2018 at 2:06

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Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of this paper of Steve Lack). The solution to your troubling corollary is that the "correct" models for weak $\omega$-categories are not the weak complicial sets but the saturated weak complicial sets, i.e. those weak complicial sets in which all the equivalences are marked. In the Street nerve of a strict $\omega$-category only the identities are marked, and it is thus not saturated in general (unless the $\omega$-category has no non-identity equivalences, in which case all pseudofunctors into it are strict).

The saturated weak complicial sets are indeed the fibrant objects of a model structure on the category of stratified simplicial sets, which is a localisation of the model structure whose fibrant objects are the weak complicial sets. See Emily Riehl's lecture notes Complicial sets, an overture, in particular Example 3.3.5.

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  • $\begingroup$ As expected! Thank you very much! I assume that the saturation condition is what I put in my comment. I will check out Emily's paper! $\endgroup$
    – Harry Gindi
    Commented May 3, 2018 at 4:28
  • $\begingroup$ @HarryGindi (and AC) What's an example of a strict complicial set which isn't saturated? (BTW, I think that instead of "saturated", one should say either "Rezk-complete" or "univalent".) $\endgroup$
    – Tim Campion
    Commented May 3, 2018 at 16:32
  • $\begingroup$ @TimCampion I mentioned one earlier: $N_{Street}(Σ(G_2))$, the stratified street nerve of the two-point suspension of the contractible groupoid on two objects. It's not saturated because all of its two-simplices represent equivalences but they aren't marked. $\endgroup$
    – Harry Gindi
    Commented May 3, 2018 at 17:34
  • $\begingroup$ The maps in question are closely relsted to the maps you localize at for $\Theta$-sets and spaces. $\endgroup$
    – Harry Gindi
    Commented May 3, 2018 at 17:42
  • $\begingroup$ Right, the similarity to other localizations is why I prefer a more suggestive name than "saturated". I'm going to have to process the following fact: If I localize a strict complicial set to be a saturated weak complicial set, and then localize again to get back to strict complicial sets, I don't get back what I started with! Although thinness in degrees $\geq 2$ is analogous in the strict and weak world (equivalence in both cases), thinness in degree 1 is modeling different notions in the weak world (where it's equivalence) and the strict world (where it's identity). $\endgroup$
    – Tim Campion
    Commented May 3, 2018 at 18:05

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