Understanding the adjunction $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightleftharpoons \mathbf{Cat}_\Delta:\mathcal{N}$

Question

Let $\mathbf{Cat}_\Delta$ denote the category of simplicially enriched categories (ie simplicial objects in $\mathbf{Cat}$ with all face and degeneracy maps bijective on objects). There is a canonical functor $\mathfrak{C}:\Delta\rightarrow\mathbf{Cat_\Delta}$ that takes $[n]$ to the simplicial category $\mathfrak{C}[n]$ defined as follows:

For $i, j\in[n]$, let $P_{ij}$ be the poset with objects $J\subseteq[n]$ containing $i$ as a least element and $j$ as a greatest element. (In particular $P_{ij}$ is empty if $i>j$.) Then $\mathfrak{C}[n]$ has objects the set $[n]$, and mapping spaces $\text{Map}_{\mathfrak{C}[n]}(i, j)$ given by the nerve of $P_{ij}$. The morphisms

       $\text{Map}_{\mathfrak{C}[n]}(i_0, i_1)\times \text{Map}_{\mathfrak{C}[n]}(i_1, i_2)\times \cdots \times \text{Map}_{\mathfrak{C}[n]}(i_{k-1}, i_k)\rightarrow \text{Map}_{\mathfrak{C}[n]}(i_0, i_k)$

are induced by the poset maps $P_{i_0 i_1}\times\cdots P_{i_{k-1} i_k}\rightarrow P_{i_0 i_k}$ given by $(J_0, \cdots, J_k)\mapsto J_0\cup\cdots\cup J_k$.

Finally, morphisms in $\Delta$ are mapped to morphisms in $\mathbf{Cat}_\Delta$ in the obvious way. We can now define a functor (the "homotopy coherent nerve") $\mathcal{N}:\mathbf{Cat_\Delta}\rightarrow \mathbf{Set}^{\Delta^{op}}$ as follows: for $\mathcal{C}$ a simplicially enriched category, let the $n$-simplices of $\mathcal{N}(\mathcal{C})$ be the set $\text{Hom}_{\mathbf{Cat_\Delta}}(\mathfrak{C}[n], \mathcal{C})$. Face and degeneracy maps are given by post-composition with the co-face and co-degeneracy morphisms on the $\mathfrak{C}[n]$ in $\mathbf{Cat_\Delta}$, and $\mathcal{N}(f)$ is given by pre-composition with $f$ for any morphism $f$ in $\mathbf{Cat_\Delta}$.

This functor is not too difficult to wrap one's head around; essentially the $n$-simplices of $\mathcal{N}(\mathcal{C})$ can be thought of as "homotopy-coherent diagrams" in $\mathcal{C}$. For instance, $0$-simplices are simply objects of $\mathcal{C}$, and $1$-simplices are vertices of the mapping spaces of $\mathcal{C}$ (ie "morphisms" of $\mathcal{C}$). $2$-simplices are specified by a trio $x, y, z$ of objects of $\mathcal{C}$, a pair of vertices (ie "morphisms") $f\in \text{Map}_\mathcal{C}(x, y)_0$ and $g\in \text{Map}_\mathcal{C}(y, z)_0$, and a 1-simplex $\sigma\in \text{Map}_\mathcal{C}(x, z)_1$ with one of its faces the composition $g\circ f$. $\sigma$ can be thought of as giving a "homotopy" from $g\circ f$ to some other morphism in $\text{Map}_\mathcal{C}(x, z)$. This idea generalizes naturally for higher values of $n$.

The issue for me now is in defining a left adjoint to $\mathcal{N}$. This is done is a purely formal way by extending the functor $\mathfrak{C}:\Delta\rightarrow\mathbf{Cat}_\Delta$. Indeed, every simplicial set $S$ is a colimit of a small diagram of $\Delta^i$s in $\mathbf{Set}^{\Delta^{op}}$, and the category $\mathbf{Cat}_\Delta$ admits small colimits, so define $\mathfrak{C}:\mathbf{Set}^{\Delta^{op}}\rightarrow\mathbf{Cat}_\Delta$ by letting $\mathfrak{C}(S)$ be the colimit of the corresponding diagram of $\mathfrak{C}[i]$s. It is immediate by construction that $\mathfrak{C}$ and $\mathcal{N}$ will be adjoint.

While I understand this on a formal level, I am having an enormous amount of difficult visualizing what the functor $\mathfrak{C}$ actually "looks like". In particular I am not able to follow computations involving $\mathfrak{C}$ at all. Does anyone know of a way of better understanding this functor on an intuitive level? I've tried to give a more explicit construction of a general $\mathfrak{C}(S)$ but have thus far failed; clearly it should have as its objects the vertices of $S$, but I have come up short in trying to describe its mapping spaces and composition laws. I know that morally $\mathfrak{C}$ should be the "free simplicial category generated by $S$", but it is really unclear to me how this should look. Any insight or references would be greatly appreciated, thank you so much.

(Also was not sure whether to post this here or on math.stackexchange; if it's better suited for the latter site feel free to close and I will repost there.)

Edit: in light of the comments below perhaps the right question to ask is this; what are some simplicial sets $S$ for which it will be particularly illuminating to compute $\mathfrak{C}(S)$ explicitly? (Rather than trying to do a general computation.)

Answer 1

To a fairly crude approximation: $\newcommand{\C}{\mathfrak{C}}$think of the functor $\C(-)$ as like geometric realisation, but realising the basic simplices as the categories $\C[n]$ instead of as the topological simplices. Lots of functors out of $\hat{\Delta}$ are defined analogously by left Kan extension of some functor on $\Delta$, and I find thinking of such functors as “roughly like geometric realisation” is always a good first approximation.

Of course, this depends on first understanding the categories $\C[n]$ — but you should be already described how to do that, since you describe the simplices appearing in $\mathcal{N}(-)$ as “not too difficult to wrap one’s head around […] homotopy coherent diagrams”, and the categories $\C[n]$ are just the categories that represent such diagrams, i.e. $\C[n]$ consists of essentially just a “homotopy coherent $n$-diagram” and nothing more.

The other thing that can be helpful for concrete computations is noticing that the functor $\C(-)$ preserves cofibrations — in particular, it maps the the boundary inclusions of simplicial sets to injective-on-objects functors, and colimits of such functors are fairly well-behaved in $\mathrm{Cat}$. So given a small combinatorial simplical sets, write it out explicitly as a “cell complex”, i.e. a composite of pushouts of the boundary inclusions; then the realisation functor will take this to a corresponding “cell complex” in $\mathrm{Cat}$, which should typically give a clear combinatorial description of the realisation.