# Examples and definition of cocompact objects

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?

• You mean a cofiltered limit. – Qiaochu Yuan Jul 17 '14 at 0:44
• 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 $\mathbf{Pro}(\mathcal{A})$. – Zhen Lin Jul 17 '14 at 7:09
• Thanks for the comments. Do you know of any examples or non-examples in $Sets$ or $Top$? – user8463524 Jul 17 '14 at 9:52
• I believe the only cocompact objects in $\mathbf{Set}$ are $0$ and $1$. – Zhen Lin Jul 17 '14 at 10:04