# Cofibrations in diagram category

Let $$\mathcal{C}$$ be a model category and $$\mathcal{I}$$ ba a small category. Then we have the projective model category structure on the diagram category $$\mathcal{C}^\mathcal{I}$$ where fibrations and weak equivalences are given point-wise fibrations and weak equivalences. That is we call a morphism $$F: D \to D'$$ in $$\mathcal{C}^\mathcal{I}$$ fibration( weak equivalence) if $$F(i): D(i) \to D'(i)$$ is fibration( weak equivalence) for all $$i \in \mathcal{I}.$$ Therefore, cofibrations are those who have left lifting property with respect to trivial fibrations.

We know from Theorem 6.36 of $$\mathit{Modern \; Classical\; Homotopy\; Theory}$$ by Strom, that a diagram $$D: \mathcal{I} \to \mathcal{C}$$ is cofibrant if and only if the canonical map $$colim D_{ is a cofibration for all $$i \in \mathcal{I}.$$ For example the diagram $$\begin{array}{ccc} A & \xrightarrow{} & B \\ \downarrow & & \downarrow \\ C & \xrightarrow{} & D\end{array}$$ is cofibrant in $$\mathcal{C}^\mathcal{I}$$ if $$A$$ is cofibrant in $$C$$, $$A\to B$$, $$A\to C$$ are cofibrations in $$\mathcal{C}$$ and finally the map $$P \to D$$ is also a cofibration where $$P$$ is the pushout of $$C \leftarrow A \to B.$$

Question: Let $$F : D \to D'$$ be a map in $$\mathcal{C}^\mathcal{I}$$. When it'll be a cofibration in $$\mathcal{C}^\mathcal{I}$$?

One condition should be the map $$F(i): D(i) \to D'(i)$$ is a cofibration for all $$i \in \mathcal{I}.$$

## 1 Answer

Strom's Theorem 6.36 requires $$I$$ to be simple, which in his terminology means that I is a poset that admits a conservative functor to the poset N of natural numbers.

This means that $$I$$ is a Reedy category with $$I_+=I$$ and the projective model structure on $$I$$-indexed diagrams coincides with the Reedy model structure.

Cofibrations in the Reedy model structure are easy to describe: these are precisely the morphisms for which all relative latching maps are cofibrations.

In our case, the latching map of a diagram $$D$$ at object $$i$$ is the canonical map $$l_i D\colon L_i D=\mathop{\rm colim} D_{ and the relative latching map of a natural transformation $$D→D'$$ is the canonical map $$L_i D' ⊔_{L_i D} D(i)→D'(i).$$

• Thank you so much for the beautiful answer. Apr 29, 2020 at 22:03