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In ordinary categories $\mathcal{C}$, there are nice conditions under which a generator is also a colimit-generator for $\mathcal{C}$. In other words, under suitable conditions, if $\mathcal{C}$ contains a family of objects $X_{\alpha}$ such that for all $C \in \mathcal{C}$ there exists an effective epimorphism $\coprod X_{\alpha} \to C$, then the smallest cocomplete subcategory of $\mathcal{C}$ containing all the $X_{\alpha}$'s is $\mathcal{C}$ itself. (See the notes https://home.sandiego.edu/~shulman/papers/generators.pdf by Mike Shulman)

I'm wondering if something similar holds in higher categories, where "higher category" could mean anything from a $2$-category to an $\infty$-category. Since we have a notion of effective epimorphism in higher categories, we can also say what it means for a collection of objects to generate the category. Under what conditions will these objects generate the category under (iterated) colimits?

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    $\begingroup$ I asked a related question once and my impression is that these sorts of questions are generally open. Note, though, that $\infty$-categorically an effective epimorphism is generally not an epimorphism, despite the name. $\endgroup$
    – Tim Campion
    Commented Nov 7, 2019 at 21:47
  • $\begingroup$ The proof of Brown representability shows that it suffices that the "generator" merely detect isomorphisms, if it's made up of compact objects. To see something more like your claim, it seems like one needs a better understanding of when regular epimorphisms are strong, that is, left orthogonal to monomorphisms. This does happen in $\infty$-toposes, at least. $\endgroup$
    – Kevin Carlson
    Commented Nov 12, 2019 at 2:00

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