Lax monoidal fibrant replacement for marked simplicial sets

Question

The category $\mathrm{Set}_{\Delta}^{+}$ of marked simplicial sets has a model structure (the (co)cartesian model structure) constructed by Lurie in HTT.3.1.3. I would like to know if this model structure admits a fibrant replacement functor $R$ that is lax monoidal with respect to the cartesian product of marked simplicial sets, that is, for which there are natural maps (necessarily equivalences) $$ R(X) \times R(Y) \to R(X \times Y), $$ which commute with the fibrant replacement maps coming from $X \times Y$ and which are associative and unital in the usual sense.

Answer 1

Not a complete answer. The following is inspired by Dmitri Pavlov's answer here.

Let's recall the work of Dugger and Spivak. Let $\mathcal G$ be a "category of gadgets closed under wedges" (Definition 5.4) (a full subcategory of $sSet_{\ast,\ast}$ satisfying certain conditions). Dugger and Spivak define a functor

$$\mathfrak C^{\mathcal G}: sSet \to sCat$$

$$Ob \mathfrak C^{\mathcal G}(S) = S_0 \qquad \qquad Hom_{\mathfrak C^{\mathcal G}(S)}(a,b) = N(\mathcal G \downarrow S)_{a,b}$$

where $N$ is the ordinary nerve, $\downarrow$ is the ordinary over-category, and the subscript means we restrict to have the correct endpoints. They show (Thm 5.2) that $\mathfrak C^{\mathcal G}$ is connected by a zigzag of natural DK equivalences to the usual functor $\mathfrak C$ (adjoint to the homotopy coherent nerve $\mathcal N$).

Moreover, the map $\theta: \mathfrak C^{\mathcal G}(X) \times \mathfrak C^{\mathcal G}(Y) \to \mathfrak C^{\mathcal G}(X \times Y)$ of Proposition 6.2 appears to make $\mathfrak C^{\mathcal G}$ into a lax monoidal functor! It is defined on homspaces by taking binary products (a category of gadgets is required to be closed under finite products). Be warned -- I have not checked this carefully! If there's a difficulty, then it probably lies in having to choose binary products in a coherent way...

Assuming this is true, the composite functor $R = \mathcal N Ex^\infty_\ast \mathfrak C^{\mathcal G}$ is a composite of lax monoidal functors and so is lax monoidal. It takes values in quasicategories, but unfortunately it is only connected by a zigzag of natural weak equivalences to the identity.

Also, this is all in the setting of the Joyal model structure -- I'm not sure about adapting it to the marked case.

EDIT: Although Dugger and Spivak don't say it, I believe there is a natural transformation $\mathfrak C \Rightarrow \mathfrak C^{\mathcal G}$, i.e. $1 \Rightarrow \mathcal N \mathfrak C^{\mathcal G}$. It sends a simplex $\sigma \in X_n$ to the simplicial functor $\mathfrak C \Delta^n \to C^{\mathcal G} X$ which does the obvious thing on objects, sends the "free" 1-morphism $f_{ij}$ from $i$ to $j$ to the composite $\Delta^{j-i} \to \Delta^n \xrightarrow \sigma X$ (where the face $\Delta^{j-i} \subseteq \Delta^n$ is the one whose long edge is the edge from $i$ to $j$), sends the homotopy $f_{jk} f_{ij} \to f_{ik}$ to the obvious map $\Delta^{k-j} \vee \Delta^{j-i} \to \Delta^{k-i}$, and is otherwise determined by the equality between the two composite maps $\Delta^{l-k} \vee \Delta^{k-j} \vee \Delta^{j-i} \to \Delta^{l-i}$ and 2-coskeletality (the homspaces of $\mathfrak C^{\mathcal G} X$ are nerves of categories and therefore 2-coskeletal).

With this transformation $1 \Rightarrow \mathcal N \mathfrak C^{\mathcal G} \Rightarrow R = \mathcal N Ex^\infty_\ast \mathfrak C^\mathcal{G}$ (assuming it's a levelwise weak equivalence), we have a fibrant replacement in the usual sense. However, the natural transformation $1 \Rightarrow R$ is not monoidal. (It does appear to be monoidal up to a natural homotopy defined using diagonals, which can probably be made as coherent as you want, but you'd have to be careful about what that means...)