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.
Patrick Nicodemus (Aug 10 2023 at 21:18):There are at least two notions of pointwise left Kan extension for the category of double categories. There is a notion due to Grandis and Pare which is conceptually similar to pasting comma squares in a 2-category, this is developed in their paper "Lax kan extensions for double categories" , and we can also consider the virtual equipment of weak double categories, pseudo double functors and (lax) double profunctors, and apply a notion of weighted colimit (which i first found in @Nathanael Arkor's paper on formal category theory)
Is it known whether these agree where they are both defined? my instinct is that the weighted colimit notion might be something closer to what Grandis and Pare call a one dimensional horizontal colimit.
Patrick Nicodemus (Aug 10 2023 at 21:25):in my application all the double categories are strict and even flat so i don't need a lot of generality
Nathanael Arkor (Aug 10 2023 at 21:33):My impression was that the notion of Kan extension for double categories studied in Lax Kan extensions for double categories was an instance of the notion of Kan extension in a double category studied in Kan extensions in double categories. If this is the case, then it may be possible to use the results of @Roald Koudenburg's On pointwise Kan extensions in double categories (e.g. Theorem 5.11 and the following corollaries) to prove that the two notions of Kan extension (in the Grandis–Paré sense, or the weighted colimit sense) coincide.
Patrick Nicodemus (Aug 11 2023 at 01:18):Ok I thought I had a response to this but I will look at it more carefully first.