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An object $X$ of a locally small category $C$ that admits filtered colimits is called compact if $$ \operatorname{Hom}_{C}(X,-) $$ preserves filtered colimits.

Let $C$ be a locally small category that admits filtered limits. Let's define an object $X$ of $C$ to be cocompact if $$ \operatorname{Hom}_{C}(-,X) $$ sends cofiltered limits to filtered colimits in $Sets$.

I never heard of this definition before. Is it reasonable? If yes, what are typical examples of cocompact objects in $Sets$, $Top$, etc?

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    $\begingroup$ You mean a cofiltered limit. $\endgroup$ – Qiaochu Yuan Jul 17 '14 at 0:44
  • $\begingroup$ Cocompactness is not as useful as compactness because many categories of interest are $\mathbf{Ind}(\mathcal{A})$ for some $\mathcal{A}$, but not so many are $\mathbf{Pro}(\mathcal{A})$. $\endgroup$ – Zhen Lin Jul 17 '14 at 7:09
  • $\begingroup$ Thanks for the comments. Do you know of any examples or non-examples in $Sets$ or $Top$? $\endgroup$ – user8463524 Jul 17 '14 at 9:52
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    $\begingroup$ I believe the only cocompact objects in $\mathbf{Set}$ are $0$ and $1$. $\endgroup$ – Zhen Lin Jul 17 '14 at 10:04

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