Joins of, and limits in, $(\infty,1)$-categories via profunctors

Question

I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.

In the classical setting it is almost a triviality to express the join of two categories as a collage along the terminal profunctor: given $\cal C,D$ and the profunctor $\varphi\colon \cal C\to D$ sending all $(C,D)$ to the terminal object in $Set$, the join ${\cal C}\star{\cal D}$ can be written as $ {\cal C}\uplus_\varphi{\cal D}$, where I denote $\uplus_\varphi$ the collage operation.

Now I wonder if, choosing simplicially enriched categories as a model instead of the quasicategorical one, I can describe the join of two $(\infty,1)$-categories $\cal C,D\in{\bf sSet}\text{-}{\bf Cat}$ to be the collage ${\cal C}\uplus_\varphi{\cal D}$ along the terminal $\bf sSet$-profunctor sending any two objects $(C,D)$ to the terminal simplicial set.

Joins of $\infty$-categories are particularly useful to define limits and colimits of diagrams $p\colon K\to\cal C$ to an $(\infty,1)$-category: following T.1.2.13.4 one can define the limit and colimit of $p$ respectively to be the terminal and initial objects in the $(\infty,1)$-categories $\hom_p(\Delta^\bullet\star K,\cal C)$ and $\hom_p(K\star\Delta^\bullet,\cal C)$.

I wonder if this point of view can be generalized to include weighted limits/colimits in $(\infty,1)$-categorical language: here is the question.

Consider the simplicial category $\mathfrak C[\Delta^n]$, for any $n$; given a simplicial functor $p\colon\cal K\to C$ between simplicial categories, how can I figure out the join (in the sense described before) $\mathfrak C[\Delta^n]\star \cal K$, and the (simplicial) set of simplicial functors $\hom_p(\mathfrak C[\Delta^n]\star \cal K,\cal C)$ which equal $p$ when restricted to $\cal K$? What is its terminal object, when it exists?

I also wonder if it is possible to encode weighted limit theory using collages along arbitrary profunctors, and then translate also this theory in the $(\infty,1)$-categorical setting. Given a generic $\bf sSet$-profunctor $W\colon \mathfrak C[\Delta^n]\to \cal K$, what should the collage $\mathfrak C[\Delta^n]\uplus_W\cal K$ be? What should the terminal object in $\hom_p(\mathfrak C[\Delta^n]\uplus_W\cal K, C)$ be?

Last but not least:

Does the "classical" definition of join between simplicial sets $K\star S$ coincide with the coherent nerve of the simplicial category $\bf K\star S$ obtained as the collage along the terminal $\bf sSet$-profunctor? This would automatically entail that the coherent nerve in $\star$-monoidal, as it is known to be.

Edit: It is maybe useful to give an idea about where I began wondering about this: all stemmed from reading this page, which I was trying to complete.

Answer 1

I am not sure I can say anything useful about the weighted limit question; I want to remark about the other two however.

The join of simplicial (properly speaking, simplicially enriched, which I still denote as $sCat$) categories is indeed preserved by the coherent nerve construction. However, the functor $\mathfrak C$ is not monoidal. For instance, we observe that $\mathfrak C[\Delta^1]=\Delta^1$ joined with itself in $sCat$ is $\Delta^3$, not $\mathfrak C[\Delta^3]$. The right statement is that for two simplicial sets, the natural map $\mathfrak C[S \star T] \to \mathfrak C[S] \star \mathfrak C[T]$ is an equivalence of simplicial categories (but not in general an isomorphism). This all is mentioned in HTT, after Remark 1.2.8.2.

In relation to the discussion about (homotopy) limits, it is important to observe that the join $\star: sCat \times sCat \to sCat$ is not monoidal for the Bergner model structure (my example above with $\mathfrak C [\Delta^1]$ is an instance of that: $\Delta^3$ is not cofibrant). In particular, looking at simplicial sets of the form $sCat(\mathcal{D} \star \mathfrak C[ \Delta^\bullet ], \cal C)$ is not a good idea as (besides the fact we don't see it is a quasicategory) the domain is not necessarily cofibrant and hence the thing we get might have incorrect homotopical behaviour.

We might replace $\mathcal{D} \star \mathfrak C[ \Delta^\bullet ]$ by $\mathfrak C[N\mathcal{D} \star \Delta^\bullet ]$ and then $$sCat(\mathfrak C[N\mathcal{D} \star \Delta^\bullet ], \mathcal C) \cong sSet (N\mathcal D \star \Delta^\bullet, N \mathcal C)$$ which (after taking fibres over the map $p: \mathcal D \to \mathcal C$ of interest) is the familiar cocone quasicategory which is used to define a colimit of $Np$ (and T.4.2.4.1 tells us the relation between quasicategorical and homotopy (co)limits).

To recap, the problem I see with the simplicial categorical join (the collage along the terminal profunctor) is that it does not interact well with the model structure. As the join of simplicial sets has correct homotopical behaviour (T.4.2.1.3), the category $Ho(QCat)$ and hence $Ho(sCat)$ do have a derived monoidal join operation, but it is not clear if there is a way to properly represent it on $sCat$. One might have more luck doing this in a different cartesian model for $(\infty,1)$-categories (like Segal categories for injective model sturcture).