Homotopy invariance of Kan nerve of simplicial categories

Question

The following question concerns the well-known paper of Dwyer and Kan "Localization of Simplicial Categories". They define a nerve for simplicial categories (with fixed set of objects $O$), by the following construction: given $\mathcal{A}\in sO-Cat$, they set $$\mathcal{N}\mathcal{A}:=diag(k \mapsto N \mathcal{A}_k)$$ where $N:Cat \to sSet$ is the usual nerve, and $\mathcal{A}_k$ is obtained by $\mathcal{A}$ by considering only the $k$-simplices in each mapping space.

Why does it hold that a weak equivalence $F:\mathcal{A} \to \mathcal{B}$ in $sO-Cat$ (i.e. a simplicial functor inducing weak equivalences $\mathcal{A}(X,Y) \to \mathcal{B}(FX,FY)$ for any $X,Y \in \mathcal{A}$) should give a weak equivalence $$\mathcal{N} \mathcal{A} \to \mathcal{N} \mathcal{B} \ \ \ ?$$ Of course this would follow by proving that we have weak equivalences $$N\mathcal{A}_k \to N\mathcal{B}_k$$ for any $k \geq 0$, but I do not get why it should be the case.

Thanks in advance for any help of hint.

Answer 1

Let $\mathcal{N}_*\mathcal{A}$ denote the bisimplicial set $k \mapsto N\mathcal{A}_k$. If $\mathcal{A} \to \mathcal{B}$ is a DK-equivalence, then $\mathcal{N}_*\mathcal{A} \to \mathcal{N}_*\mathcal{B}$ is a natural weak equivalence of bisimplicial sets, except not in $k$ but rather in $l$ the "nerve simplicial direction". This can be easily seen by realizing that $\mathcal{N}_*\mathcal{A}$ can be also written as

$$(\mathcal{N}_*\mathcal{A})_l = \coprod_{x_0, \ldots, x_l \in O} \mathcal{A}(x_0, x_1) \times \ldots \times \mathcal{A}(x_{l-1}, x_l) \text{.}$$