# The left adjoint functor $\mathfrak{C}$ to the simplicial nerve functor $N$.

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 $$\operatorname{Set}_{\Delta}$$?

$$\mathfrak C[S]$$ 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 $$\Delta^n$$.
• Sorry, I feel a little bit confused with the definition of $\mathfrak C[\Delta^n]$, does that mean the objects are from $\{0,1,...,n\}$, with Hom(i,j)= the set $\{I\subset [n] : (i, j\in I)\wedge (∀k \in I)[i ≤ k ≤ j]\}$? – Danny Sep 30 '18 at 21:20
• (if $i\le j$ and $\emptyset$ if $i>j$) – Danny Sep 30 '18 at 21:27