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Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ containing $S$ is all of $\mathcal{C}$. Under what conditions is it true that for every object $x \in \mathcal{C}$, there exists a functor $f: \mathcal{I} \to \mathcal{C}$ whose image is contained in $S$ and whose colimit is isomorphic to $x$? This is true for example if $\mathcal{C}$ is a presheaf category and $S$ the canonical set of generators (the representables). I suspect the answer is no in general, but I don't know how to construct an example.

I am also interested in the analogous question for presentable $\infty$-categories.

Edit: After posting this question, I became aware of Mike Shulman's answer here which addresses the same question with the counterexample being the category of compact Hausdorff spaces. But is there a natural presentable example that one might expect to come across (some type of structured sets, for instance)? I'd like to get some intuition for this phenomenon.

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  • $\begingroup$ Every locally presentable $\infty$-category is also a locally presentable 1-category, so the counterexample mentioned below by Todd also applies to that case. $\endgroup$
    – Mike Shulman
    Commented Jan 26, 2015 at 5:53

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Here is a short note by Mike Shulman with some relevant material. He remarks that the arrow category $\mathbb{2}$ inside $Cat$ (which of course is presentable) is a colimit generator in your sense (see his definition 3.6), but it is not a colimit-dense generator (see his definition 3.5) in the sense that any object of $C$ is a colimit of a functor into the full subcategory of $Cat$ containing $\mathbb{2}$. So this answers at least an implicit question of the post.

I think I'll give him a poke, because he's undoubtedly better placed to handle this query.

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  • $\begingroup$ Interesting. Could you (or Mike) perhaps elaborate why $\mathbb{2}$ is not a colimit-dense generator in $Cat$? $\endgroup$
    – Akhil Mathew
    Commented Jan 26, 2015 at 3:23
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    $\begingroup$ @AkhilMathew, it's roughly the same reason why the one-point space is not colimit-dense in CptHaus. Cat is monadic over reflexive graphs, 2 is free on a reflexive graph, and its endomorphisms as a category are the same as those of that graph. Thus, any diagram in the full subcategory on 2 is the image of a diagram of reflexive graphs, hence its colimit is also free on a reflexive graph. But not every category is free. (More intuitively, the problem is that such a colimit cannot introduce equations between composites.) $\endgroup$
    – Mike Shulman
    Commented Jan 26, 2015 at 5:52
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    $\begingroup$ Todd and @MikeShulman, thanks for explaining this. $\endgroup$
    – Akhil Mathew
    Commented Jan 26, 2015 at 14:00
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    $\begingroup$ @Mike: in your note you claim that $\mathbb{Z}$ is colimit-dense in $\text{Ab}$. I don't see why this is true, and in fact I suspect it's false, although I can't prove it. The problem is that a colimit of copies of $\mathbb{Z}$ corresponds to a presentation of an abelian group by generators and relations, but where the only relations we're allowed to impose are that one generator is a multiple of another. I don't know how to give, say, the $p$-adic integers a presentation of this form. $\endgroup$
    – Qiaochu Yuan
    Commented May 1, 2015 at 10:03
  • $\begingroup$ @QiaochuYuan I guess that you are right. I thought that I got that statement from somewhere else, but right now I can't find it anywhere, and your objection seems valid. Can you think of an example of a colimit-dense generator that is not dense? $\endgroup$
    – Mike Shulman
    Commented Aug 28, 2015 at 17:46

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