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Ivan Di Liberti (Dec 05 2025 at 16:15):I do not think there is a name for this notion. I think I would suggest a name like incremented, or appended. Let me explain the intuition, and I would be very happy to discuss different names.
An existing related notion is that of cellular categories. While there is no formal definition of cellular colimit, morally a construction is cellular if it formed by taking directed colimits or pushouts. Now, your situation is the one in which you are not entirely allows to glue stuff, you can only glue along the empty set, which is what I would call append.
James Deikun (Dec 05 2025 at 16:25):Hm, does this end up being the same as the class of categories where every finite connected diagram has a cone?
Tom Hirschowitz (Dec 05 2025 at 16:41):@James Deikun Do you mean the obtained colimits raher than the categories?
@Ivan Di Liberti Thanks for the suggestion! I had thought of syntactic, because this is the kind of colimit we use when computing initial endofunctor algebras.
James Deikun (Dec 05 2025 at 18:56):Yes, of course the set of categories is not literally the same, but I think the set of obtained colimits could be. In which case, "componentwise filtered", maybe.
Kevin Carlson (Dec 05 2025 at 19:07):Yes, I was also going to say that this seems like a rather unnatural class, but if you informally take the pushout to get categories whose connected components, that's much more natural. I don't know a short name for this class, although locally within a paper I would probably use "locally filtered", but it is at least quite important in that it's the class of colimits commuting with finite connected limits, equivalently with pullbacks, in Set.
Kevin Carlson (Dec 05 2025 at 19:10):According to the terminology of A classification of accessible categories, you could call these the "FINCL-filtered categories".
Kevin Carlson (Dec 05 2025 at 19:12):James Deikun said:
Hm, does this end up being the same as the class of categories where every finite connected diagram has a cone?
And, yeah, once you give the generalization you suggested, James, this is right, see the same citation.
Nathanael Arkor (Dec 05 2025 at 21:02):These are called "pseudo-filtered categories/colimits" (see Notes on Commutation of Limits and Colimits).
Tom Hirschowitz (Dec 05 2025 at 22:44):Great, thanks to you all for the help and pointers!
Notification Bot (Dec 05 2025 at 22:45):Tom Hirschowitz has marked this topic as resolved.