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While the $\alpha$-filtered colimit idea does not work, Brown representability can be proved efficiently for these categories. Use the fact that an $\alpha$-presentable $\infty$-category is reflective in a finitely presentable one, and steal Brown representability from there--it's easy to prove that a reflective subcategory of a category on which every functor preserving coproducts and weak pushouts is representable enjoys the same properties. When the resulting finitely presentable $\infty$-category is generated by a set of compact cogroups, you win. Unfortunately, as discussed above, the homotopy categories of finitely presentable $\infty$-categories do not usually have Brown representability available to steal. There is a way out: the spheres do detect isomorphisms in the homotopy 2-category of the $\infty$-category of spaces, and from there it follows that the whole Brown representability story goes through perfectly for all $\alpha$-presentable $\infty$-categories--the only cost is that you have to take functors valued in groupoids, not in sets. Your mileage may vary on how "citeable" this is, but see 5.3.6 in my 2020 thesis, Some 2-categorical aspects of $\infty$-category theory.

While the $\alpha$-filtered colimit idea does not work, Brown representability can be proved efficiently for these categories. Use the fact that an $\alpha$-presentable $\infty$-category is reflective in a finitely presentable one, and steal Brown representability from there--it's easy to prove that a reflective subcategory of a category on which every functor preserving coproducts and weak pushouts is representable enjoys the same properties. When the resulting finitely presentable $\infty$-category is generated by a set of compact cogroups, you win. Unfortunately, as discussed above, the homotopy categories of finitely presentable $\infty$-categories do not usually have Brown representability available to steal. There is a way out: the spheres do detect isomorphisms in the homotopy 2-category of the $\infty$-category of spaces, and from there it follows that the whole Brown representability story goes through perfectly for all $\alpha$-presentable $\infty$-categories--the only cost is that you have to take functors valued in groupoids, not in sets. Your mileage may vary on how "citeable" this is, but see 5.3.6 in my 2020 thesis, 2-categorical Brown representability and the relation between derivators and $\infty$-categories.

While the $\alpha$-filtered colimit idea does not work, Brown representability can be proved efficiently for these categories. Use the fact that an $\alpha$-presentable $\infty$-category is reflective in a finitely presentable one, and steal Brown representability from there--it's easy to prove that a reflective subcategory of a category on which every functor preserving coproducts and weak pushouts is representable enjoys the same properties. When the resulting finitely presentable $\infty$-category is generated by a set of compact cogroups, you win. Unfortunately, as discussed above, the homotopy categories of finitely presentable $\infty$-categories do not usually have Brown representability available to steal. There is a way out: the spheres do detect isomorphisms in the homotopy 2-category of the $\infty$-category of spaces, and from there it follows that the whole Brown representability story goes through perfectly for all $\alpha$-presentable $\infty$-categories--the only cost is that you have to take functors valued in groupoids, not in sets. Your mileage may vary on how "citeable" this is, but see 5.3.6 in my 2020 thesis, Some 2-categorical aspects of $\infty$-category theory.

While the $\alpha$-filtered colimit idea does not work, Brown representability can be proved efficiently for these categories. Use the fact that an $\alpha$-presentable $\infty$-category is reflective in a finitely presentable one, and steal Brown representability from there--it's easy to prove that a reflective subcategory of a category on which every functor preserving coproducts and weak pushouts is representable enjoys the same properties. When the resulting finitely presentable $\infty$-category is generated by a set of compact cogroups, you win. Unfortunately, as discussed above, the homotopy categories of finitely presentable $\infty$-categories do not usually have Brown representability available to steal. There is a way out: the spheres do detect isomorphisms in the homotopy 2-category of the $\infty$-category of spaces, and from there it follows that the whole Brown representability story goes through perfectly for all $\alpha$-presentable $\infty$-categories--the only cost is that you have to take functors valued in groupoids, not in sets. Your mileage may vary on how "citeable" this is, but see 5.3.6 in my 2020 thesis, Some 2-categorical aspects of $\infty$-category theory.

The above framework is valid in any locally presentable $\infty$-category $\mathcal C$, and heuristically one should get counterexamples in essentially any such $\infty$-category and for essentially any choices of $Z$ and $(X_i)$. Franke's argument (introduction of On the Brown representability theorem for triangulated categories, Topology 40 (2001) pp.667-680) for this heuristic in the stable situation was as follows: There is a spectral sequence converging to $\mathrm{Hom}_{\mathcal C}(\mathrm{colim} X_i,Z)$ whose second page contains $\lim^q H^p(\mathrm{Hom}_{\mathcal C}(X_i,Z))$, and the functor represented by $Z$ will only send $\mathrm{colim} X_i$ to a weak limit of sets if this spectral sequence collapses. Unfortunately, nobody seems to know how to compute these derived limits well enough to show that what intuitively ought to happen actually does happen-there are isolated explicit examples of ordinal-indexed sequences of abelian groups with nontrivial derived limits, but I don't know how to cook one into this spectral sequence.

The above framework is valid in any locally presentable $\infty$-category $\mathcal C$, and heuristically one should get counterexamples in essentially any such $\infty$-category and for essentially any choices of $Z$ and $(X_i)$. Franke's argument (introduction of On the Brown representability theorem for triangulated categories, Topology 40 (2001) pp.667-680, doi:10.1016/S0040-9383(99)00034-8) for this heuristic in the stable situation was as follows: There is a spectral sequence converging to $\mathrm{Hom}_{\mathcal C}(\mathrm{colim} X_i,Z)$ whose second page contains $\lim^q H^p(\mathrm{Hom}_{\mathcal C}(X_i,Z))$, and the functor represented by $Z$ will only send $\mathrm{colim} X_i$ to a weak limit of sets if this spectral sequence collapses. Unfortunately, nobody seems to know how to compute these derived limits well enough to show that what intuitively ought to happen actually does happen-there are isolated explicit examples of ordinal-indexed sequences of abelian groups with nontrivial derived limits, but I don't know how to cook one into this spectral sequence.