# Start a new discussion

## Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeSep 7th 2012

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.

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeJan 31st 2014
• (edited Jan 31st 2014)

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!

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJan 31st 2014

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.

• CommentRowNumber4.
• CommentAuthorTim_Porter
• CommentTimeJan 31st 2014

Thanks Todd. You confirmed what I had thought was in fact correct.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 31st 2014

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeJan 31st 2014

I tried to say something extra at filtered category.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 31st 2014

Thanks! Excellent.

• CommentRowNumber8.
• CommentAuthorTim_Porter
• CommentTimeFeb 1st 2014

I second that!