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I added a section to filtered category about generalized filteredness relative to a class of small categories, as studied by Adamek-Borceux-Lack-Rosicky, and mentioned that it yields a better notion of $\kappa$-filteredness for the finite regular cardinal $2$, as pointed out by Zhen in another thread.
Can I ask clarification on the sentence:
The usual filtered categories are then the case $\kappa = \omega$.
I had though that $\kappa$ was a regular cardinal, whilst $\omega$ was an ordinal. Was $\aleph_0$ intended or have I missed something, (in which case I would suggest that an additional word or two would be useful). My reason for asking was that I have been looking for the precise relationship between compact object and $\kappa$-compact object, and the entries in the Lab do not give the relationship in simple terms (i.e. simple enough for me :-(). I presume ‘compact = $\aleph_0$-compact’. The question seems to hinge, as well, on whether certain < are $\leq$ or not!
Right, cardinals are by definition (in most approaches used today) certain types of ordinals, and $\omega$ happens to be that type of ordinal. It would have been okay to say $\aleph_0$-compact objects, but it’s much more usual to see reference in the literature to $\omega$-compact objects, which are the same as compact objects.
The condition for a category $C$ to be $\kappa$-filtered is that for every diagram $D \to C$ where $D$ is of size $\lt \kappa$, there is an extension $D^+ \to C$ where $D^+$ is obtained by adjoining a terminal object to $D$. So $\omega$-filtered means that every diagram in $C$ of size $\lt \omega$, i.e., every finite diagram $D \to C$, has such an extension; these are called just filtered categories. And so $\omega$-compact has to do with $hom(c, -)$ preserving filtered colimits.
Thanks Todd. You confirmed what I had thought was in fact correct.
Since the entry apparently didn’t make this clear enough: Todd, might you have a minute to add some more explanation to the entry? So that the next reader will know for sure? That would be great.
I tried to say something extra at filtered category.
Thanks! Excellent.
I second that!
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