In HTT, a version of the adjoint functor theorem for (locally) presentable infinity categories is proven (Corollary 5.5.2.9). Is there a more refined version of this somewhere, which more closely resembles Freyd’s original version? I.e., is there a version for infinity categories which are not necessarily (locally) presentable, but requires a solution set condition? If so, a reference would be fantastic. Thanks!
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1$\begingroup$ Perhaps one can work this out by oneself? After all, the adjoint functor has always a canonical representation as a colimit/limit, and the sole purpose of the solution set condition is to provide a small cofinal subcategory so that the colimit/limit exists. So then one has to look for the appropriate notions for $\infty$-categories. $\endgroup$– Martin BrandenburgCommented Jun 19, 2012 at 9:18
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$\begingroup$ I agree with you in principle Martin. However, I still haven't figured out how to prove this question (mathoverflow.net/questions/97562/…) in the infinity context, and the classical way to prove the adjoint functor theorem uses this result for 1-categories. Any ideas? $\endgroup$– David CarchediCommented Jun 19, 2012 at 17:07
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The general adjoint functor theorem didn't exist in the literature when this question was originally asked, but now it does: Nguyen, Raptis, and Schrade.
answered Aug 8, 2018 at 14:38