I am reading "A model structure on the category of small categories for coverings " https://arxiv.org/abs/0907.5339 and am interested in the question that "What is a cylinder object in ${\bf Cat}_{1}$?".
Define a model structure on the category ${\bf Cat}$ of small categories as follows:
(1) A functor $f : \mathcal{C} \to \mathcal{D}$ is a cofibration if and only if the map $Ob(f) : Ob(\mathcal{C}) \to Ob(\mathcal{D})$ between sets of objects is injective.
(2)A functor $f : \mathcal{C} \to \mathcal{D}$ is a weak equivalence if and only if for each object $x \in Ob(\mathcal{C})$, if the induced maps $\pi_{0}(f) : \pi_{0}(\mathcal{C}) \to \pi_{0}(\mathcal{D})$ and $\pi_{1}(\mathcal{C}, x) \to \pi_{1}(\mathcal{D}, f(x))$ are isomorphisms.
(3) fibrations are defined as functors that have the right lifting property with respect to acyclic cofibrations.
The category ${\bf Cat}$ of small categories equipped with the above model structure is denoted by ${\bf Cat}_{1}$.
I expect that $\mathcal{C} \times J_{m}$ is a cylinder object in ${\bf Cat}_{1}$ for any small category $\mathcal{C}$ where $J_{m}$ is a poset described as follows:
$J_{m} : 0 \to 1 \leftarrow 2 \to 3 \leftarrow 4 \to \cdots (\leftarrow) \to m$.
Yes, this is true. This follows inductively from the fact that all categories $\mathcal C$ are cofibrant and that $\mathcal C \times J_1$ is a cylinder object for $\mathcal C$. For the inductive step I will use a similar method to the proof of a proposition in this text on model categories. The proposition in question is proposition $1.2.5$ (iii) found on page 9.
I will summarize it here but for the details the text is called "Model Categories" and is written by Mark Hovey.
They use the fact that if $B'$ and $B''$ are cylinder objects for $B$ and $H':B' \rightarrow X$ and $H'':B'' \rightarrow X$ are left homotopies $f \rightarrow g$ and $g \rightarrow h$ respectively then we can create a new cylinder object $B_*$ by the pushout
Now letting $\mathcal C = B$, $B' = B'' = \mathcal C \times J_1$ we see that if we choose $i_0 :\mathcal C \rightarrow \mathcal C \times J_1$ and $i_1: \mathcal C \rightarrow \mathcal C \times J_1$ to be (in coordinates) $id_{\mathcal C}$ and the constant functor onto $0$ and $1$ respectively and $j_0 = i_1$ and $j_1 = i_0$ we see that $(B'',j)$ and $(B',i)$ are cylinder objects for $\mathcal C$.
What is $B_*$ in this scenario then? Well $\mathcal C \times -$ is a left adjoint since $\textbf{Cat}$ is a cartesian closed category so the resulting pushout commutes with products, i.e $B_* = \mathcal C \times D $. Where $D$ is the pushout
$*$ denoting the category with one object and one morphism and $0$ and $1$ denoting the constant functors onto those object. Clearly $D = J_2$ which shows that $\mathcal C \times J_2$ is a cylinder object for $\mathcal C$. We can continue the process inductively to show that $J_{n+1}$ is a pushout of $J_n$, $J_1$ and $*$.
Hope this was helpful!