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Given a cocomplete category $C$, is there an example of an object which is small but not compact?

I am working with the following definitions of small and compact:

Given a cardinal $\kappa$ one says that an object $X$ is $\kappa$-compact, if ${\rm Hom}(X,-)$ commutes with $\kappa$-filtered colimits. One says $X$ is $\kappa$-small if the same happens if indexing category of the colimit is an ordinal.

small means $\kappa$-small for some $\kappa$

compact means $\kappa$-compact for some $\kappa$

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    $\begingroup$ See Exercise 1.c(4) in "Locally presentable and acessible categories", Adámek & Rosicky. $\endgroup$ Commented Oct 22, 2015 at 21:36

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There is no difference for $\kappa = \aleph_0$. The point is that you can build colimits for filtered diagrams using just colimits for chains.

  1. Every filtered category $\mathcal{J}$ admits a cofinal directed diagram, i.e. a cofinal functor $\mathcal{I} \to \mathcal{J}$ where $\mathcal{I}$ is directed.
  2. Every countable directed poset $\mathcal{I}$ admits a cofinal $\omega$-chain: just take an enumeration of the elements of $\mathcal{I}$ and repeatedly use directedness to get a cofinal chain of length $\omega$.
  3. Every directed poset $\mathcal{I}$ of cardinality $\lambda$ is the union of a $\lambda$-chain of directed subposets of cardinality $< \lambda$. (Observe that every infinite subset $S \subseteq \mathcal{I}$ is contained in a directed subposet of $\mathcal{I}$ of the same cardinality as $S$.)

Thus, by induction, every directed diagram in $\mathcal{C}$ has a colimit constructed using only colimits for chains.

It is tempting to try to generalise this to regular cardinals $\kappa > \aleph_0$, but the subtlety is in (3): in general, $\kappa < \lambda$ is not enough to imply that every subset $S$ of a $\kappa$-directed poset $\mathcal{I}$ of cardinality $< \lambda$ is contained in a $\kappa$-directed subposet of $\mathcal{I}$ of cardinality $< \lambda$. (For this, we need $\kappa \triangleleft \lambda$; see Theorem 2.11 in [Locally presentable and accessible categories].)

I suppose the point is that, for the purposes of the small object argument, $\kappa$-smallness suffices. But one often gets $\kappa$-compactness as well.

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    $\begingroup$ This shows what goes wrong if you try to prove that they are equivalent. But it would still be good to see a specific counterexample, as the question asks for! $\endgroup$ Commented Dec 3, 2014 at 16:52
  • $\begingroup$ It would be good to know if $\kappa$-smallness and $\kappa$-compactness really are different. But given that $\aleph_1 \not\triangleleft \aleph_{\omega + 1}$ is basically the smallest non-example of $\kappa \triangleleft \lambda$, it's probably going to be quite intricate... $\endgroup$ Commented Dec 3, 2014 at 17:42
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    $\begingroup$ Thanks Zhen for your answer. Another issue with generalising 3. is that $\lambda$ that appears may not be $\kappa$-filtered. does one face similar difficulty if one tries to prove $\kappa$-smallness is $\gamma$-compactness for some large enough cardinal $\gamma$? $\endgroup$ Commented Dec 4, 2014 at 1:22
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    $\begingroup$ I would think so. But that claim is even harder to disprove because it is vacuous in a locally presentable category. $\endgroup$ Commented Dec 4, 2014 at 7:34

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