# When is $Ind(\mathcal{C})$ equivalent to $PSh(\mathcal{C})$?

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?

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