# A model structure on marked simplicial sets

Do you have a reference for the following fact? And before that, is it true?

The Joyal model structure on simplicial sets "lifts" to a model structure on the category of marked simplicial sets, having as fibrant objects precisely the objects sent to fibrant objects by the obvious forgetful functor $\mathbf{sSet}^+\to\mathbf{sSet}$.

If it is not true, what should be a best approximation to it and where can I find it?

Thanks!

• The fibrant objects do not determine the model structure. But they do if you specify what you want the cofibrations to be. Even so, I'm puzzled. What's wrong with the usual model structure, where $X$ is fibrant if its underlying simplicial set is a quasicategory and the marked morphisms are precisely the equivalences (and the cofibrations are the monomorphisms)? Aug 27 '17 at 0:56

The existence of the model structure on $$\operatorname{Set}_{\Delta}^{+}$$ that you are most probably looking for can be deduced by employing Corollary 3.3.4 of A necessary and sufficient condition for induced model structures by Kathryn Hess, Magdalena Kędziorek, Emily Riehl and Brooke Shipley. It is called the right-induced model structure (a.k.a. transferred model structure) with respect to the forgetful functor $$\operatorname{Set}_{\Delta}^{+} \to \operatorname{Set}_{\Delta}$$. The forgetful functor $$\operatorname{Set}_{\Delta}^{+} \to \operatorname{Set}_{\Delta}$$ preserves and reflects both weak equivalences and fibrations.

The weakly saturated class of cofibrations in $$\operatorname{Set}_{\Delta}^{+}$$ is generated by the inclusions of the form $$(\partial \Delta^n)^{\flat} \to (\Delta^n)^{\flat}$$, $$n \in \mathbb{N}_0$$, where $$\cdot\,^{\flat} : \operatorname{Set}_{\Delta} \to \operatorname{Set}_{\Delta}^{+}$$ is the left adjoint to the forgetful functor $$\operatorname{Set}_{\Delta}^{+} \to \operatorname{Set}_{\Delta}$$.

• Beware that in the transferred model structure, not every monomorphism is a cofibration. Aug 27 '17 at 0:58
• @TimCampion There is no model structure in which every monomorphism is a cofibration and fibrant objects are those specified by OP. The reason is that marked simplices in fibrant objects must be stable under homotopy in this case. But there is a model structure in which cofibrations are precisely monomorphisms and a marked simplicial set is fibrant if and only if it is a quasicategory and marked simplices are stable under homotopy. Aug 27 '17 at 11:03
• @user62782 That's interesting. Can you give a reference where that model structure is constructed, i.e. its existence is proved? Aug 27 '17 at 16:30
• @DanielGerigk This follows from the general construction in arxiv.org/abs/1610.08459 (which is quite straightforward). It applies only to a larger category of marked simplicial sets that you talked about in addendum II, but the model structure can be transferred to the smaller category then. Aug 27 '17 at 17:05
• @user62782 Yes. Perhaps there is even a model structure on $\operatorname{Cat}^{+}$, where a cofibration is a functor that is injective on objects ( - as in the canonical model structure on $\operatorname{Cat}$ - ), weak equivalences are the ones you described, and a marked category is fibrant if and only if the property of being marked of a morphism in that category is invariant under isomorphism. Aug 28 '17 at 13:21

I'm still unclear on what the motivation for looking at such a model structure might be, so I'm going to go ahead and guess that what you're ultimately interested in, Fosco, is an alternate construction of the usual model structure on $\mathrm{Set}_\Delta^+$ constructed in HTT 3.1.3.7 (where we take $S$ to be a point). Apologies if I'm guessing wrong!

Verity's approach:

Verity almost does this. To see this, recall that a stratified simplicial set in Verity's sense is a simplicial set equipped with certain "thin" simplicies which can have arbitrary dimension $\geq 1$ (and it's required that every degenerate simplex is thin). Verity constructs a model structure on the category $\mathrm{Strat}$ of stratified simplicial sets whose fibrant objects are what Verity calls weak complicial sets. A marked simplicial set can be considered as a stratified simplicial set by taking a 1-simplex to be thin iff it is marked, and taking all simplices of dimension $\geq 2$ to be thin. Then Verity's model structure on $\mathrm{Strat}$ restricts to a model structure on $\mathrm{Set}_\Delta^+$.

The only caveat is that Verity's model structure needs to be further localized, by a "Rezk completeness condition". That is, in his model structure, every fibrant marked simplicial set will have the property that every marked simplex is an equivalence, but will not necessarily have the converse property that every equivalence is marked. This needs to be enforced by a Bousfield localization. This is not hard to do, but I'm not sure it's been done in the literature.

The Cisinski/Olschok approach:

It's notable that Verity's construction of his model structure is very much analogous to Joyal's construction of the Joyal model structure, which in turn is essentially an application of Cisinski's theory of model structures on Grothendieck topoi. The only reason that Cisinski's theory can't be directly applied to marked simplicial sets is that marked simplicial sets don't actually form a topos! But that's okay -- Olschok has extended Cisinski's theory to general locally presentable categories. Olshok's theory is a very general method for constructing model structures from a set of generating cofibrations, a set of elementary anodyne extensions, and a functorial cylinder object.

In order to apply Olschok's theory, we must simply identify a class of elementary anodyne extensions and a functorial cylinder object (it's easy to write down a set of generating cofibrations such that the cofibrations are exactly the monomorphisms). The elementary anodyne extensions from Verity's paper (supplemented by a morphism corresponding to Rezk completeness), and the functorial cylinder given by $X \mapsto X \times \Delta[1]_t$ (where $\Delta[1]_t$ is the marked 1-simplex) yield the usual model structure. Olschok's theory allows us to identify the fibrant objects and fibrations between fibrant objects in this model structure, via lifting against the elementary anodyne extensions, if we additionally verify that the anodyne extensions are closed under certain pushouts involving the functorial cylinder; This is a straightforward combinatorial exercise.