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I have a question about the injective model structure on functor categories.

As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are at least two model structures on the functor category $[\mathcal{C}, \mathcal{M}]$. These are the injective and the projective model structures that share the same weak equivalences, the morphisms that are obejct-wise weak equivalences in $\mathcal{M}$. More precisely,

In the injective : the cofibrations are the object-wise cofibrations, while the fibrations are the maps with the right lifting property with respect to acyclic cofibrations (i.e., object-wise acyclic cofibrations).

In the projective : the fibrations are the object-wise fibrations, while the cofibrations are the maps with the leftlifting property with respect to acyclic fibrations (i.e., object-wise acyclic fibrations).

These are important models, they corespond for example to the injective and projective models on chain complexes or on simplicial (pre)sheaves.

Both structures are cofibrantly generated. In the projective model, the generating cofibrations may be given by $ \coprod_{\mathcal{C}(C,-)}{i} \colon \coprod_{\mathcal{C}(C,-)}{A} \to \coprod_{\mathcal{C}(C,-)}{B}$ where $i \colon A \to B$ is a generating cofibration in $\mathcal{M}$ and $C \in \mathcal{C}$. The injective version is obscure for me.

My question is : What can be taken as generating (acyclic) cofibrations in the injective models ? Even a particular case such as chain complexes is appreciated.

Thanks, Bogdan

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An explicit construction of generating (acyclic) cofibrations in the injective model structure can be found after Remark A.3.3.14 in Lurie's Higher Topos Theory. These are all injective (acyclic) cofibrations whose source and target is (componentwise) κ-compact for some appropriately chosen cardinal κ. –  Dmitri Pavlov Oct 16 at 12:54

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