While reading Rezk's paper "A model for the homotopy theory of homotopy theory", I found a remark which contradicts a guess of mine, but I can't see where I am wrong (perhaps it might be a silly mistake, though I can't see where the following reasoning fails).


Given a category $\mathscr{C}$ and a subcategory $\mathscr{W}\subset \mathscr{C}$ such that $ob(\mathscr{W})=ob(\mathscr{C})$, we say that an arrow $f$ in $\mathscr{C}$ is a weak equivalence if it belongs to $\mathscr{W}$. We then define the simplicial space $N(\mathscr{C},\mathscr{W})$, which is, in dimension $m$, given by $\text{nerve} ( we(\mathscr{C}^{[m]}))$, where $we(\mathscr{C}^{[m]})$ is the subcategory of the functor category $\mathscr{C}^{[m]}$ spanned by pointwise weak equivalences.


We now have two canonical examples: the discrete nerve, where $\mathscr{W}$ is given by the identities (i.e. it is the underlying discrete category), and the classifying diagram $\textbf{N}\mathscr{C}$, where $\mathscr{W}$ is given by the isomorphisms in $\mathscr{C}$, i.e. it is the maximal subgroupoid of $\mathscr{C}$.


It seems to me that the former is a subcase of the latter, since $\text{discnerve}(\mathscr{C})=\textbf{N}\tilde{\mathscr{C}}$, where $\tilde{\mathscr{C}}$ is obtained from $\mathscr{C}$ by retaining only the identities. But Rezk proves $\textbf{N}\mathscr{C}$ to be always a complete Segal space (for $\mathscr{C}$ explicitly small, though I think it is always understood to be so, otherwise there are some problems in the definition), while he then says that $\text{discnerve}(\mathscr{C})$ is not always such.


Where am I wrong? Thanks in advance



Is the discrete nerve of a small category a complete Segal space?
Your assertion $discnerve(\mathcal{C})=N\tilde{\mathcal{C}}$ is false.
Indeed, the category $id(\mathcal{C}^{[m]})$ (identities between chains of $m$ maps of $\mathcal{C}$), whose nerve is $discnerve(\mathcal{C})\_m$, and the category $\tilde{\mathcal{C}}^{[m]}$
(identities between chains of $m$ maps of $\tilde{\mathcal{C}}$), whose nerve is $N\tilde{\mathcal{C}}$, have not the same objects.


