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Stream: practice: terminology & notation

Topic: Notation for Kan extensions and liftings

Mike Shulman (Jan 18 2023 at 16:34):

Does anyone know who first used the notations $\mathrm{Lan}$ and $\mathrm{Ran}$ for left and right Kan extensions, respectively?

Mike Shulman (Jan 18 2023 at 16:35):

And relatedly, has anyone seen a halfway-standard similar notation for Kan liftings?

Evan Patterson (Jan 18 2023 at 16:57):

I've seen people use $\mathrm{Rift}$ for right Kan lifts, which seems suggestive and memorable.

Nathanael Arkor (Jan 18 2023 at 16:57):

I'm not sure if it's the earliest reference, but, as an upper bound, Lan and Ran appear in Dubuc's Kan Extensions in Enriched Category Theory (1970).

Nathanael Arkor (Jan 18 2023 at 16:58):

And I have seen Lift and Rift used for left and right Kan lifts respectively, though I wouldn't know where to look for an early reference.

Nathanael Arkor (Jan 18 2023 at 18:42):

Nathanael Arkor said:

I'm not sure if it's the earliest reference, but, as an upper bound, Lan and Ran appear in Dubuc's Kan Extensions in Enriched Category Theory (1970).

Looking around a little more, it doesn't seem entirely unlikely to me that this is the first occurrence of the terminology. Prior papers like Day–Kelly's Enriched functor categories (1969) don't give a specific name to the Kan extensions (other than some letter).

Mike Shulman (Jan 18 2023 at 19:00):

That sounds plausible.