# Cofibrant simplicial categories

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?

• Indeed, any free category is cofibrant as a (locally discrete) simplicial category. Where do you see a suggestion to the contrary? (Note also that $[2]$ is not isomorphic to $\mathfrak{C}(\Delta[2])$, since the hom-space Hom($0$,$2$) in the latter is isomorphic to $\Delta[1]$.) – Alexander Campbell Nov 13 '19 at 11:01
• Of course, my mistake, I'll edit and shift every number down by 1 – Giulio Lo Monaco Nov 13 '19 at 13:03
• It was really more of a feeling: since in almost every source where the structure of $\mathfrak{C}$ is described the authors seem to care about telling me that it provides a cofibrant replacement functor for the discrete simplicial categories of the form $[n]$, I've been naturally led to believe that we actually need to replace them in order to get something cofibrant. – Giulio Lo Monaco Nov 13 '19 at 13:53
• As pointed out in Valery's answer, both $[n]$ and $\mathfrak{C}\Delta^n$ are cofibrant simplicial categories. The key point here is that both of these actually describe functors $\Delta\to \mathrm{sCat}$, but only the second of these functors is cofibrant as an object of $Fun(\Delta, \mathrm{sCat})$ (where cofibrant means "Reedy cofibrant"). – Charles Rezk Nov 13 '19 at 16:33

Simplicial categories $$[n]$$ are indeed cofibrant. The Simplicial category $$\mathfrak{C}\Delta^n$$ is also cofibrant and weakly equivalent to $$[n]$$, so the former is a cofibrant replacement of the latter. Note that $$\mathfrak{C}\Delta^n$$ is isomorphic to $$[n]$$ only when $$n \leq 1$$ (so this is not true for $$n = 2$$). The point is that simplicial functors $$\mathfrak{C}\Delta^n \to C$$ carry more information than $$[n] \to C$$, so the homotopy coherent nerve of $$C$$ remembers enough information about $$C$$ as opposed to the ordinary nerve.