In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve:

Definition Let $\mathcal{C}$ be a simplicial category. The simplicial nerve $N(\mathcal{C})$ is the simplicial set described by the formula

$$ \text{Hom}_{\mathcal{S}et_{\Delta}}(\Delta^n,N(\mathcal{C}))=\text{Hom}_{\mathcal{C}at_{\Delta}}(\mathfrak{C}[\Delta^n],\mathcal{C}). $$

Lurie then goes on to say (in Warning that if one considers a simplicial category $\mathcal{C}$ as an ordinary category by forgetting about all the positive dimensional simplices in the mapping spaces of $\mathcal{C}$, then (generally) the simplicial nerve of $\mathcal{C}$ will not coincide with the ordinary notion of the nerve of $\mathcal{C}$.

Question: Why is this the case? What exactly is the difference between the two?

I mean, it seems clear to me that the difference should lie in the forgetting of simplices in $\mathcal{C}$ when considering it as an ordinary category, which would still be present in the case of the simplicial nerve, but I'm just trying to get a better sort of intuitive picture of the differences between the two. Anyone have a nice example that highlights this?

  • $\begingroup$ Again, the difference should lie in the fact that the simplicial nerve is a higher-dimensional construction (as evinced by, for example, Prop., which shows that if the mapping spaces are all Kan complexes, then the simplicial nerve is an $\infty$-category), but I was just wondering if anyone has a nice, simple, specific example that highlights the differences. $\endgroup$ – Ralph Mellish Feb 6 '14 at 2:05

Take, for example, a simplicial category $\mathcal{C}$ with exactly one object $*$ and $B G$ for its unique simplicial set of morphisms, where $B G$ is the classifying Kan complex for an abelian group $G$. (Composition is induced by the group structure of $G$.) Then:

  • As an ordinary category, $\mathcal{C}$ is isomorphic to the terminal category $\mathbb{1}$.
  • As a simplicial category, $\mathcal{C}$ is Dwyer–Kan equivalent to $\mathbb{1}$ if and only if $G$ is the trivial group.

Since the simplicial nerve is conservative up to the appropriate notion of weak equivalence, we deduce that the simplicial nerve of $\mathcal{C}$ cannot be contractible.


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