# model structure on the arrow category

I wish to show that given a model structure on a category $C$ one can define a model structure on $C^2$, where $2$ denotes the category with two objects and only one non-identity morphism.

I define the fibrations (respectively cofibrations, weak-equivalences) of $C^2$ to be the morphisms which are fibrations "objectwise", that is natural transformation $\tau$ such that $\tau_0$ and $\tau_1$ are both fibrations (respectively cofibrations, weak-equivalences) in model category $C$ we started with.

I want to show that that $Cof \subset LLP(Fib \cap WE)$. My idea is to do it "objectwise": given a commuting diagram in $C^2$ one gets two commuting diagram in $C$ by evaluating at $0$ and $1$. Because of the definition of $Fib$, $Cof$ and $WE$ we obtain a lift in each degree.

My only issue is showing that the natural transformation we get from those two lifts is indeed natural.

## 1 Answer

It's not natural, in general! There are two natural model structures on $C^2$, the "projective" and the "injective." Both have the objectwise weak equivalences. The projective fibrations are objectwise, and dually for the injective cofibrations, but the projective cofibrations and injective fibrations are more complicated.

Suppose I have a square $\begin{matrix} f&\to &g\\\downarrow &&\downarrow\\ h&\to&k\end{matrix}$ in $C^2$ such that $g\to k$ is a trivial projective fibration, that is, an objectwise trivial fibration. To construct a lift $h\to g$, we must be able to choose a lift $h_0\to g_0$. So $f_0\to g_0$ must be a cofibration in $C$-so far, so good. It's similarly necessary that $f_1\to g_1$ be a cofibration in $C$, but it's not sufficient. To come up with a sufficient condition, suppose we've chosen a lift $h_0\to g_0$. Then the composed map $h_0\to g_1$, combined with the given map $f_1\to g_1$, produce a map from the pushout $h_0\sqcup_{f_0} f_1\to g_1$ such that the square $\begin{matrix} h_0\sqcup_{f_0} f_1&\to &g_1\\\downarrow &&\downarrow\\ h_1&\to&k_1\end{matrix}$ commutes in $C$. So if the natural map $h_0\sqcup_{f_0} f_1\to h_1$ is a cofibration in $C$, then the original map $f\to h$ had the left lifting property with respect to $g\to k$, and we've found a sufficient condition. In fact, this is also necessary, and so the projective model structure has for its cofibrations the maps $f\to h$ such that $f_0\to h_0$ and $h_0\sqcup_{f_0} f_1\to h_1$ are cofibrations in $C$, and dually for the injective structure.

You would probably find it instructive to try to fill in the details of this yourself. But if you want a reference, this is part of a far more general story of model structures on functor categories indexed by direct and inverse categories, and more generally still, on those indexed by Reedy categories. You can find some more details on these things on the nLab, and full proofs in the standard references, namely the books of Hovey and Hirschhorn.