I need some help understanding the working of the homotopy coherent nerve (as described in Lurie's HTT).
Let $i < j$, then $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) \simeq (\Delta^1)^{(j-i-1)}$, and for any $i < j$ with $(i,j) \neq (0,n)$ we have $\mathrm{Hom}_{\mathfrak{C}\Delta^n}(i, j) =\mathrm{Hom}_{\mathfrak{C}\partial\Delta^n}(i, j)$.
Lurie states that $\mathrm{Hom}_{\mathfrak{C}\partial\Delta^n}(0, n) = \partial(\Delta^1)^{n-1}$. Why is this so?
Update: I'm now semi-convinced (having computed it by hand on some examples), however I'm having problems proving it. Any help?
Let $n \in \mathbb{N}_{\geq 1}$. The simplicial category $\mathfrak{C} [\partial \Delta^n]$ has as objects the set $\{0,\ldots,n\}$, and the simplicial set $Hom_{\mathfrak{C}[ \partial \Delta^n]}(i,j)$, for $0 \leq i < j \leq n$ and $(i,j) \neq (0,n)$, is $N(P_{i,j})$, the nerve of the partially ordered set of subsets $I \subseteq \{ i,\ldots,j \}$ with $i,j \in I$. The simplicial set of morphisms $Hom_{\mathfrak{C} [\partial \Delta^n]}(0,n)$ is the union over all chains of objects $0 = x_0 < \ldots < x_l = n$ of the image of the map $N(P_{x_0,x_1}) \times \ldots \times N(P_{x_{l-1},x_l}) \rightarrow N(P_{0,n}) = Hom_{\mathfrak{C} [\Delta^n]}(0,n)$. You'll find that a $k$-simplex of $Hom_{\mathfrak{C} [\Delta^n]}(0,n)$ is a chain $I_0 \subseteq I_1 \subseteq \ldots \subseteq I_k \subseteq \{ 0,\ldots,n \}$ with $0,n \in I_0$, and this $k$-simplex is in $Hom_{\mathfrak{C} [\partial \Delta^n]}(0,n)$ if and only if either $I_0 \neq \{ 0,n \}$ or $I_k \neq \{ 0,1,\ldots,n \}$. If you regard $Hom_{\mathfrak{C} [\Delta^n]}(0,n)$ as a box $(\Delta^1)^{n-1}$, then $Hom_{\mathfrak{C}[\partial \Delta^n]}(0,n)$ corresponds precisely to the "boundary" of that box.