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I know the general definition for simplicial nerve $\mathfrak{C}:Set_{\Delta}\rightarrow Cat_{\Delta}$.We define what are $\mathfrak{C}[\Delta^n]$ and $\mathfrak{C}[f]$ for morphisms between those $\Delta^n$'s.Just as HTT says:

Definition 1.1.5.1. Let $J$ be a finite nonempty linearly ordered set. The simplicial category $\mathfrak{C}\left[\Delta^{J}\right]$ is defined as follows:

  • The objects of $\mathfrak{C}\left[\Delta^{J}\right]$ are the elements of $J$.
  • If $i, j \in J$, then $$ \operatorname{Map}_{\mathfrak{C}\left[\Delta^{J}\right]}(i, j)=\left\{\begin{array}{ll} \emptyset & \text { if } j<i \\ \mathrm{~N}\left(P_{i, j}\right) & \text { if } i \leq j \end{array}\right. $$ Here $P_{i, j}$ denotes the partially ordered set $\{I \subseteq J:(i, j \in I) \wedge(\forall k \in$ I) $[i \leq k \leq j]\}$.
  • If $i_{0} \leq i_{1} \leq \cdots \leq i_{n}$, then the composition $$ \operatorname{Map}_{\mathfrak{C}\left[\Delta^{J}\right]}\left(i_{0}, i_{1}\right) \times \cdots \times \operatorname{Map}_{\mathfrak{C}\left[\Delta^{J}\right]}\left(i_{n-1}, i_{n}\right) \rightarrow \operatorname{Map}_{\mathfrak{C}\left[\Delta^{J}\right]}\left(i_{0}, i_{n}\right) $$ is induced by the map of partially ordered sets $$ \begin{array}{c} P_{i_{0}, i_{1}} \times \cdots \times P_{i_{n-1}, i_{n}} \rightarrow P_{i_{0}, i_{n}} \\ \left(I_{1}, \ldots, I_{n}\right) \mapsto I_{1} \cup \cdots \cup I_{n} . \end{array} $$

Then we extend this definition by requiring $\mathfrak{C}$ persevering colimits.

My question is how we explicitly calculate $\mathfrak{C}[S]$ for a simplicial set $S$.I played with two examples $Map_{\mathfrak{C}[\Lambda_1^3]}(0,3)$ and $Map_{\mathfrak{C}[\Lambda_1^4]}(0,4)$:

$Map_{\mathfrak{C}[\Lambda_1^3]}(0,3)$

Here $Map_{\mathfrak{C}[\Lambda_1^3]}(0,3)$ consists of 4 0-simplexes,3 1-simplexes.

$Map_{\mathfrak{C}[\Lambda_1^4]}(0,4)$

Here $Map_{\mathfrak{C}[\Lambda_1^4]}(0,4)$ is obtained by deleting all 3-simplexes forming the interior of the cube $Map_{\mathfrak{C}[\Delta^4]}(0,4)$ and two 2 simplexes forming one face of $Map_{\mathfrak{C}[\Delta^4]}(0,4)$.

Are those calculations right?

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    $\begingroup$ This is a great exercise! I think that first picture looks more like $Map_{\mathfrak C[\Lambda^3_2]}(0,3)$ than $Map_{\mathfrak C[\Lambda^3_1]}(0,3)$? EDIT: Nope, actually I agree with your picture. $\endgroup$
    – Tim Campion
    Jan 10, 2022 at 20:33
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    $\begingroup$ yep, I think both are correct now. $\endgroup$
    – XiaYu
    Jan 11, 2022 at 5:45

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