I am referring to Lurie's book. In the following, he introduces a functor, $\mathfrak{C}$ which is left adjoint to the nerve functor $N$, which sends a simplicial set to a simplicial category, which is a category enriched over $\operatorname{Set}_{\Delta}$, the category of simplicial sets:
It is not difficult to check that the construction described in Definition 1.1.5.3 is well-defined and compatible with composition in $f$. Consequently, we deduce that $\mathfrak C$ determines a functor
$$\mathbf \Delta\to \operatorname{Cat}_{\Delta}\\\Delta^n\mapsto \mathfrak C[\Delta^n]$$
which we may view as cosimplicial object of $\operatorname{Cat}_{\Delta}$.The category $\operatorname{Cat}_{\Delta}$ of simplicial categories admits (small) colimits. Consequently, by formal nonsense, the functor $\mathfrak C : \mathbf \Delta \to \operatorname{Cat}_{\Delta}$ extends uniquely (up to unique isomorphism) to a colimit-preserving functor $\operatorname{Set}_{\Delta}\to\operatorname{Cat}_{\Delta}$, which we will denote also by $\mathfrak C$. By construction, the functor $\mathfrak C$ is left adjoint to the simplicial nerve functor $N$. For each simplicial set $S$, we can view $\mathfrak{C}[S]$ as the simplicial category "freely generated" by $S$: every $n$-simplex $\sigma: \Delta^n \to S$ determines a functor $\mathfrak C[\Delta^n]\to \mathfrak C[S]$, which we can think of as a homotopy coherent diagram $[n]\to \mathfrak C[S]$.
But I am confused with the specific definition of $\mathfrak{C}$. Even by his last sentence, I still can't see why $\mathfrak{C}[S]$ is a simplicial category: what are the objects in $\mathfrak{C}[S]$, why is $\mathfrak{C}[S]$ enriched over $\operatorname{Set}_{\Delta}$?
Best Answer
$\mathfrak C[S]$ is defined to be the colimit, among simplicial categories, of copies of $\mathfrak C[\Delta^n]$ indexed by the category of maps from simplices into $S$. This implies in particular that it’s a simplicial category, simply because these are closed under colimits. The objects of $\mathfrak C[S]$ are just the $0$-simplices of $S$. Other than by direct inspection, one way to check this quickly is to observe that the functor from simplicial set to sets giving the set of $0$-simplices factors through $\mathfrak C$, simply because it does so when restricted to the objects $\Delta^n$.