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Let $\mathcal{K}$ be a category and $\mathcal{K}_{\text{fin}}$ its free completion with finite limits.

  • Does the embedding $\mathcal{K} \hookrightarrow \mathcal{K}_{\text{fin}}$ preserve some colimits?

I am especially interested in directed colimits.

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Yes, all the colimits that exist in ${\cal K}$. Indeed, ${\cal K}_{{\rm fin}}$ can be identified with the smallest full subcategory of ${\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ which contains the representable functors ${\rm Hom}(x,-)$ and is closed under finite limits, and the composition ${\cal K} \to {\cal K}_{{\rm fin}} \hookrightarrow {\rm Fun}({\cal K},{\rm Set})^{{\rm op}}$ is the opposite of the Yoneda embedding ${\cal K}^{{\rm op}} \to {\rm Fun}({\cal K},{\rm Set})$. The result you need now follows from the fact that the Yoneda embedding preserves all limits which exist in ${\cal K}^{{\rm op}}$.

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  • $\begingroup$ Can you add some detail or reference?! For example underling why I have preservation of colimits and not limits?! $\endgroup$ – Ivan Di Liberti Oct 29 '17 at 23:02
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    $\begingroup$ You can't expect to have limits preserved in a free completion under limits, because you're adding new limits freely and in general there's no reason for the new limit of some diagram to coincide with any old limit that it used to have. (There are more refined kinds of limit completion that do preserve certain existing limits, but they aren't "free" cocompletions any more.) $\endgroup$ – Mike Shulman Oct 30 '17 at 6:28
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    $\begingroup$ @IvanDiLiberti, for a category ${\cal C}$, the Yoneda embedding ${\cal C} \to {\rm Fun}({\cal C}^{{\rm op}},{\rm Set})$ sends $x \in {\cal C}$ to ${\rm Hom}(-,x): {\cal C}^{{\rm op}} \to {\rm Set}$. Now if $x$ is the limit of a diagram $\{x_i\}_{i \in {\cal I}}$ then clearly ${\rm Hom}(-,x) = {\rm lim}_{i \in {\cal I}}{\rm Hom}(-,x_i)$, but if $x$ is the colimit of a diagram then in general there is nothing to be said about maps into $x$. That's why the Yoneda embedding preserves limits and not colimits (or why, in your case, the opposite of Yoneda preserves colimits but not limits). $\endgroup$ – Yonatan Harpaz Nov 2 '17 at 19:37

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