Let $\mathcal{C}$ be a category. We know that every presheaf $\mathcal{C}^{op} \to Set$ is a colimit over its slice category of representable functors. I'm wondering when every such colimit is a directed limit, and thus when the category $Ind(\mathcal{C})$ is equivalent to the category of presheaves on $\mathcal{C}$.

My idea for a sufficient condition: If $\mathcal{C}$ is small and has coproducts, then every colimit is directed. This is pretty obvious, since the coproduct of two objects provides the upper bound in the slice category of a presheaf, thus making it a directed set (it's a set by smallness of $\mathcal{C}$).

Is my argument correct that this is a sufficient condition for $Ind(\mathcal{C}) \cong PSh(\mathcal{C})$? If not, are there any other necessary and sufficient conditions for this to be the case?

  • $\begingroup$ If $C$ is small and has coproducts, then it's a preorder by Freyd's theorem. $\endgroup$ – Qiaochu Yuan Nov 18 '17 at 17:59
  • $\begingroup$ Wouldn't it have upper bounds as well? For any objects $A$ and $B$ you have $A \to A \coprod B$ and $B \to A \coprod B$. $\endgroup$ – leibnewtz Nov 18 '17 at 18:29
  • $\begingroup$ Okay I realized this is false. The category of finitely generated projective modules over a ring $R$ is essentially small with finite coproducts but the ind category is all modules $\endgroup$ – leibnewtz Nov 18 '17 at 19:35
  • $\begingroup$ The ind category is flat modules. $\endgroup$ – Qiaochu Yuan Nov 18 '17 at 20:19
  • $\begingroup$ Thanks. I meant to put finitely presented modules but either way it works $\endgroup$ – leibnewtz Nov 18 '17 at 22:13

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