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Assume a category $\mathcal{D}$ admits small filtrant inductive limits. Then call an object $Y$ of $\mathcal{D}$ compact, if $\hom_{\mathcal{D}}(Y,\cdot )$ commutes with these small filtrant inductive limits (i found this definition in the book "Categories and Sheaves" of M. Kashiwara and P. Schapira where this property is called finite presentation).

Now my question is:

Denoting by $\mathrm{Ind}(\mathcal{C})$ the full subcategory of Ind-objects in $\mathrm{Func}(\mathcal{C}^{\mathrm{op}},\mathrm{Set})$ and by $\mathrm{Ind}(\mathcal{C})^c$ the compact objects of $\mathrm{Ind}(\mathcal{C})$, which requirements must $\mathcal{C}$ meet to make the canonical functor $$\mathcal{C}\rightarrow \mathrm{Ind}(\mathcal{C})^c\tag{1}$$ (given by the Yoneda embedding) an equivalence of categories?

In some exercise (I think it was exercise 6.1) of the above cited book I found for example the claim that if $\mathcal{C}$ is any small category, an object $A$ of $\mathrm{Ind}(\mathcal{C})$ is compact if and only if there is an object $X$ of $\mathcal{C}$, together with morphisms $$A\overset{f}{\rightarrow}X\overset{g}{\rightarrow}A$$ in $\mathrm{Ind}(\mathcal{C})$ such that $g\circ f$ is the identity, and that hence if (and actually only if) $\mathcal{C}$ is idempotent complete, (1) is indeed an equivalence, but unfortunately I don't see why any compact object $A$ admits $X$, $f$ and $g$ as described above (the converse is rather clear). Could anyone give me a hint? And can the smallness condition on $\mathcal{C}$ be weakened or modified?

Many thanks in advance!

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The smallness condition is not especially important. The key condition is indeed idempotent-completeness.

Suppose $A$ is a compact object in $\mathbf{Ind} (\mathcal{C})$. By definition, there is a filtered diagram $Y : \mathcal{J} \to \mathcal{C}$ such that $A$ is the colimit of $Y$ in $\mathbf{Ind} (\mathcal{C})$. But the canonical comparison map $$\varinjlim \mathrm{Hom} (A, Y) \to \mathrm{Hom} (A, \varinjlim Y)$$ is a bijection, and $\varinjlim Y \cong A$, so there is some object $j$ in $\mathcal{J}$ and some morphism $f : A \to Y j$ in $\mathbf{Ind} (\mathcal{C})$ such that $g \circ f = \mathrm{id}_A$, where $g : Y j \to A$ is the component of the colimiting cocone. In particular, if $\mathcal{C}$ is idempotent-complete, then $A$ is isomorphic to some object in $\mathcal{C}$.

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  • $\begingroup$ Thank you very much for your answer! $\endgroup$ – user178979 Jun 28 '15 at 16:59

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