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We're all familiar with terminal cones/initial cocones in the form of limits/colimits.

What about initial cones and terminal cocones?

While writing an answer to a related question the concept cropped up; clearly these still have a universal property, so why do we never see them in practice? (or do we and I'm just missing it?) Some light googling revealed nothing; any pointers are appreciated.

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    $\begingroup$ If the category has an initial object or a terminal object then the initial cone or terminal cocone are just these. So in most categories where one might want to consider this type of notion, it collapses to a simpler one. $\endgroup$
    – Simon Henry
    Commented Apr 4, 2023 at 17:41
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    $\begingroup$ I think the special case of suprema illustrates this nicely: you can express these as colimits in posets, and they are interesting because they are the "smallest element larger than every element in the indexing set", but a terminal cocone would be the "smallest element smaller than every element in the indexing set", so the notion ends up reducing to just the minimal element of the poset and thus is not so interesting in general. (This is just a special case of Simon's comment, of course) $\endgroup$
    – Emily
    Commented Apr 4, 2023 at 18:10

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