Small objects vs Compact objects 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 $\kappa$
 A: There is no difference for $\kappa = \aleph_0$. The point is that you can build colimits for filtered diagrams using just colimits for chains.


*

*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.

*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 $\omega$.

*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.
