It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheaffiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

Edit: Apparently Lurie's construction has been studied further in an abstract context by Makkai and Rosický and Makkai and Rosický and Vokřínek.

It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheafifiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

Edit: Apparently Lurie's construction has been studied further in an abstract context by Makkai and Rosický and Makkai and Rosický and Vokřínek.

It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheaffiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

It seems very unlikely to me that you will be able to get any useful handle on the generating acyclic cofibrations in the injective model structure, even in simple cases like when $I$ is the delooping of a group. The only way I have ever seen to show that they exist is by using some nasty cardinality argument akin to Lurie's A3.3.3.

I believe that the first construction of the injective model structure on diagrams of simplicial sets (specifically) was in Alex Heller's monograph "Homotopy Theories", section II.4. I don't quite understand his argument at the moment; it doesn't seem to use cofibrant generation directly.

Another, somewhat more general, reference, which is also earlier than Lurie, is Tibor Beke's paper Sheafifiable homotopy model categories, which uses a logical approach and requires that the model category be not only combinatorial but "sheaffiable".

I don't think I've ever seen any construction of an injective model structure for a non-combinatorial model category.

Edit: Apparently Lurie's construction has been studied further in an abstract context by Makkai and Rosický and Makkai and Rosický and Vokřínek.