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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 $colim D_{<i} \to D(i)$ 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 $i \in \mathcal{I}.$

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

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  • $\begingroup$ Thank you so much for the beautiful answer. $\endgroup$
    – Surojit
    Apr 29, 2020 at 22:03

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