According to https://ncatlab.org/nlab/show/cellular+model, I call a cellular model of a locally presentable category a set of monomorphisms cofibrantly generating the monomorphisms. I am curious to see an example of a locally presentable category which does not have a cellular model.
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$\begingroup$ There are lots of examples and relevant observations here: mathoverflow.net/questions/209734/…. If $C$ is a locally presentable $\infty$-category then $C$ has a combinatorial model via simplicial presheaves. I think you can choose to take the injective model structure on presheaves when running that proof, and you get a cellular model. $\endgroup$– David WhiteCommented Oct 31, 2022 at 10:53
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$\begingroup$ This is tagged model categories, but clicking the nlab link, it's clear the OP's question is entirely categorical. Therefore, my previous comment is misleading and is not an answer. It shows that presentable $\infty$-categories can be modelled by cellular model categories but that's not the question. $\endgroup$– David WhiteCommented Jan 5 at 17:19
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