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Robin Jourde (he) (Oct 20 2026 at 15:59):Hello everyone,
I am looking for a theorem stating conditions for the unit of a left Kan extension in a double category (or in a 2-category) to be an isomorphism, some sort of generalisation of the pointwise+fully faithful condition (see e.g. corollary 6.3.9 in Category Theory in Context of Emily Riehl). Do you know if such a thing exists?
I've looked at Grandis&Paré's On weak double categories, parts III and IV and Koudenburg's On pointwise Kan extensions in double categories where they discuss pointwise extensions but I have not found anything about the unit being an isomorphism.
Thanks!
Nathanael Arkor (Oct 20 2026 at 21:20):What definition of left Kan extension are you using, or what is the setting in which you wish to apply such a theorem? For instance, if you are working in the setting of an equipment, this is proven in Lemma 3.29 here.
Robin Jourde (he) (Oct 21 2026 at 10:14):I am working in a 2-category of virtual double categories (with some properties, namely cartesian virtual equipments) and am wondering about the extension of some virtual double functor.
I asked about extensions in double categories because I felt like most work is done in that setting and is only more "expressive" than in 2-categories (?)
Anyway, the result you mentioned should be enough for me. Thank you for pointing that out, I'll have a read
Nathanael Arkor (Oct 21 2026 at 20:34):To talk of left extensions of virtual double categories, you really need to work in a virtual double category: VDCs do not form a double category, merely a VDC. 2-categories are also insufficient, as the representably fully faithful 1-cells in the 2-category of VDCs are not the fully faithful functors of VDCs (see §2.4 here). So, the result I linked to should indeed be the most appropriate for your purposes.