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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 -filteredness for the finite regular cardinal , as pointed out by Zhen in another thread.
Can I ask clarification on the sentence:
The usual filtered categories are then the case .
I had though that was a regular cardinal, whilst was an ordinal. Was 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 -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 = -compact’. The question seems to hinge, as well, on whether certain < are or not!
Right, cardinals are by definition (in most approaches used today) certain types of ordinals, and happens to be that type of ordinal. It would have been okay to say -compact objects, but it’s much more usual to see reference in the literature to -compact objects, which are the same as compact objects.
The condition for a category to be -filtered is that for every diagram where is of size , there is an extension where is obtained by adjoining a terminal object to . So -filtered means that every diagram in of size , i.e., every finite diagram , has such an extension; these are called just filtered categories. And so -compact has to do with 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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