Mentioned -filtered categories.
]]>Replaced a broken reference link.
]]>Removing redirects for cofiltered categories, etc, as I will create a separate page for these to spell out some details explicitly.
]]>I second that!
]]>Thanks! Excellent.
]]>I tried to say something extra at filtered category.
]]>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.
]]>Thanks Todd. You confirmed what I had thought was in fact correct.
]]>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.
]]>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!
]]>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.
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