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Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$:

  • Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := \text{Hom}_{\Delta\textsf{-Cat}}(\Delta^{\bullet},{\mathcal C})$, where $\Delta^{\bullet}$ is the cosimplicial simplicial category with $\Delta^n$ a simplicial category version of the $n$-simplex (Lurie, Def. 1.1.5.5)

  • View ${\mathcal C}$ as a category in $\textsf{sSet}$ with discrete space $X^0$ of objects, morphisms $X^1 := \bigsqcup_{x,y\in\text{Obj}({\mathcal C})}{\mathcal C}(x,y)$, and form the geometric realization of its internal nerve (a simplicial object in $\textsf{sSet}$) $$\widetilde{\text{N}}_\Delta({\mathcal C}) := \text{Real}\left(\ldots\substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow\\[-1em]\longrightarrow}X^1\times_{X^0} X^1 \substack{\longrightarrow\\[-1em] \longrightarrow \\[-1em] \longrightarrow}X^1\rightrightarrows X^0\right)$$

How can these two constructions be related? I'd also be happy if someone could provide a good reference.

Here is some vague intuition on why there might be a relation: First, in the case of an ordinary category viewed as a discrete simplicial category, both constructions agree with the ordinary nerve construction (any simplicial set is the geometric realization of itself when considered as a discrete simplicial space). Also, looking at $2$-simplices, the two constructions are similar: In the simplicial nerve $\text{N}_\Delta({\mathcal C})$, a $2$-simplex is given by morphisms $\alpha: x\to y, \beta: y\to z, \gamma: x\to z$ together with a homotopy $H: \beta\alpha\to\gamma$, so $2$-simplices in $\text{N}_\Delta({\mathcal C})$ correspond to compositions $(\alpha,\beta,\gamma,H)$ of $1$-morphisms up to homotopy. For $\widetilde{\text{N}}_\Delta({\mathcal C})$, $2$-simplices coming from the $0$-simplices of $X^1\times_{X^0}X^1$ only account for strict compositions $(\alpha,\beta,\beta\alpha,\text{id})$ in ${\mathcal C}$, but this is compensated by the fact that the $1$-simplices of $\widetilde{\text{N}}_\Delta({\mathcal C})$ coming from the $0$-simplices of $X^1$ vary 'continuously' in $X^1$.

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  • $\begingroup$ What makes you think the two constructions are related? Just because they have the same name doesn't mean they have to be the same... $\endgroup$ – Zhen Lin Jul 18 '15 at 23:48
  • $\begingroup$ @ZhenLin: I added the few thoughts that made me believe that there might be a connection. $\endgroup$ – Hanno Jul 19 '15 at 17:00
  • $\begingroup$ First things first: simplicial sets considered as objects in the Joyal model category must not be confused with simplicial sets considered as objects in the Kan–Quillen model category. The "geometric realisation" of a bisimplicial set is all about the latter, whereas the homotopy coherent nerve is all about the former. Secondly, ordinary categories are too much of a special case to tell you anything interesting here; instead you should be looking at, say, a simplicial monoid considered as a one-object simplicial category. $\endgroup$ – Zhen Lin Jul 20 '15 at 1:26

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