Reading about simplicial categories, and in particular the model structure in sCat, I found various sources, among which I can mention this one or section 16.2 in Riehl's book “Categorical homotopy theory”, which seem to imply that the trivial simplicial structures on the posets $[n]$ are not cofibrant. Now, this leaves me a bit puzzled, in that I can exhibit a very explicit description of the initial maps of these objects as compositions of pushouts of generating cofibrations. For instance, $[2]$ is the smallest of these simplicial categories which is claimed not to be cofibrant. $[1]$ is already known to be cofibrant (for one thing, it is isomorphic to $\mathfrak{C}\Delta^1$). Now consider the following two diagrams:

$\require{AMScd}$ \begin{CD} \emptyset @>>> [1]\\ @VVV @VVV\\ [0] @>>> [1] \coprod [0] \end{CD}

$\require{AMScd}$ \begin{CD} [1]_{\emptyset} @>f>> [1] \coprod [0]\\ @VVV @VVV\\ [1]_{\Delta^0} @>>> [2] \end{CD}

where $f$ selects the terminal point of $[1]$ and the adjoined point. It is clear that the first diagram is a pushout. It really seems to me that the second one is, too, both by explicit construction of a pushout in $\textbf{sCat}$ and by checking the universal property. The left vertical maps in both diagrams are generating cofibrations. I conclude that the inclusion $[1] \to [2]$ is also a cofibration. Is there a mistake that I don't see? What goes wrong in this argument?

cofibrantas an object of $Fun(\Delta, \mathrm{sCat})$ (wherecofibrantmeans "Reedy cofibrant"). $\endgroup$ – Charles Rezk Nov 13 '19 at 16:33