You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Benjamin Grant (Apr 08 2023 at 03:11):Hi folks, I’m interested to know if there’s a commonly-used name for the following property of a poset: if A and B have a common lower bound, then their least upper bound exists. This is trivially true in any join-semilattice, so I’m also interested more broadly in cases where not all joins necessarily exist. Thanks in advance!
Mike Shulman (Apr 08 2023 at 03:28):If I understand correctly, this is equivalent to saying that the poset qua category has pushouts.
Benjamin Grant (Apr 08 2023 at 05:24):I think you’re right about that!
Amar Hadzihasanovic (Apr 08 2023 at 06:41):Similarly to how pushouts are coproducts in under-slice categories, you can rephrase this property as “for all A, the upper set of A is a join-semilattice”.
I don't know if there is any established order-theoretic terminology for properties that hold “at all (principal) upper sets”.
In CT we sometimes use “local(ly)” for things that hold at all slices, but it can also mean “at all hom-sets”, and I think I've only encountered the analogue of the latter in order theory (i.e. a poset is “locally X” if every interval of the poset is X).
Benjamin Grant (Apr 08 2023 at 17:33):Makes sense to me, I think that answers my question pretty well. Thank you!