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Graham Manuell (May 15 2022 at 12:26):This is probably a naive question, but the main examples of absolute (co)limits (splittings of idempotents and biproducts) are also the main examples of 'bilimits' in the sense of being both limits and colimits. Is there some deep reason behind this?
James Deikun (May 15 2022 at 13:06):Yes! In fact all absolute (co)limits are bilimits. I think either Kelly or Street proved this in the setting of enriched categories, but I don't recall the reference. The weights for absolute colimits are the ones that have right (I think) adjoints and if you use the adjoint weight for a limit you get the same object. Similarly if you start with an absolute limit its weight has a left (I think) adjoint which gives you the same object as a colimit.
Graham Manuell (May 15 2022 at 13:36):Thanks! It looks like the paper you are referring to is Absolute colimits in enriched categories by Ross Street. The paper uses some kind of parametrised version of weighted limits that I haven't seen before, so I will probably need to take some time to wrap my head around it. Street doesn't actually mention that if you use the adjoint weight for the limit it gives the same object as the colimit, though maybe this is obvious for people who understand enriched category theory better than I do.
Martti Karvonen (May 15 2022 at 13:54):There's also Garner's Diagrammatic characterisation of enriched absolute colimits
Mike Shulman (May 15 2022 at 14:34):FYI, 'bilimit' is a potentially confusing word, because it's also been used to mean "bicategorical limit".
Graham Manuell (May 15 2022 at 16:00):Yes, though is there any other established name for this concept?
Kenji Maillard (May 15 2022 at 16:28):The nlab's [[bilimit]] page seem to propose ambidextrous adjunction as an alternative name (with both pros and cons). Looking at the arguments (and unless there exist another good disambiguation) I might use ambilimit if I need to refer to these.
John Baez (May 15 2022 at 16:59):An ambidextrous adjunction is an adjunction where the left adjoint is also a right adjoint. That's a perfectly good term for that concept. Of course that's not exactly the same as something that's both a limit and colimit. But they're connected - so yes, I like the word 'ambilimit'.
Nathanael Arkor (May 15 2022 at 23:10):See §6 of Kelly–Schmitt's Notes on enriched categories with colimits of some class, which explicitly proves the result about absolute weights in terms of adjoints.