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Bruno Gavranović (Sep 11 2022 at 00:10):Am I correct in understanding that defining a lax adjunction whose 1-cells are lax functors is still an open question?
Namely, it's been established that one can generalise the notion of an adjunction internal to a 2-category to a lax adjunction internal to a 3-category. For instance, there's a 3-category of 2-categories, 2-functors, pseudo/lax transformations and modifications where this makes sense.
But famously there isn't a 3-category where 1-cells are lax functors. There's a 1-category (of 2-categories and lax functors), a 2-category (of 2-categories, lax functors and icons), but not a 3-category.
Bruno Gavranović (Sep 11 2022 at 00:12):This seems to prohibit even talking about lax adjunctions, since there's no modifications that can be coherently defined w.r.t. lax functors.
But I wouldn't be surprised if there's a paper from the 70s shedding light on this...
Beppe Metere (Sep 11 2022 at 07:23):Hi,
Lax/colax adjunctions (in the context of double categories) are discussed in Section 3 of
Grandis Paré - Adjoint for double categories - Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 45 (2004) no. 3, pp. 193-240.
Ciao,
Beppe.
Tobias Schmude (Sep 11 2022 at 07:47):The nlab page on [[lax 2-adjunctions]] mentions two papers considering lax adjunctions with lax functors. I don't know if they yield a complete picture though, I haven't looked at the details.
Christian Williams (Sep 11 2022 at 22:34):Double categories, lax functors, lax transformations, and modifications form a tricategory. (The problem with composing lax transformations disappears when a "strict dimension" is distinguished from the lax one.)
Christian Williams (Sep 11 2022 at 22:40):You can take bicategories as vertically discrete double categories, and consider lax adjunctions in Bicat < Dbl.
Matteo Capucci (he/him) (Sep 12 2022 at 12:17):So to sum up and expand on @Christian Williams's answer (I'm doing this for myself first), there is a strict double (large) category formed by weak double categories, lax and colax functors as horizontal and vertical 1-cells respectively, and double cells as follows:
Matteo Capucci (he/him) (Sep 12 2022 at 12:20):A complete description of this double category can be found in the aforementioned paper by Grandis and Parè, Adjoint for double categories, in §2
Matteo Capucci (he/him) (Sep 12 2022 at 12:22):In their paper about intercategories, they remark:
It’s not unreasonable to wonder where the usual problems of making lax func-
tors into a 2-category have gone, given that 2-categories and bicategories can be consid-
ered as special weak double categories. In the case of 2-categories considered as horizontal
double categories, lax functors are just 2-functors and horizontal transformations are 2-
natural transformations so there are no problems there. If, on the other hand, bicategories
are considered as vertical weak double categories (i.e. horizontal arrows are identities),
then lax functors are what are usually called lax functors but horizontal transformations
are now the (dual) icons of Lack [13], whose main feature is precisely that they are the
2-cells of a 2-category.
Matteo Capucci (he/him) (Sep 12 2022 at 12:28):Then if one wants another dimension, I argue this shows they should turn to triple categories instead of tricategories. In their intercategories paper, Grandis and Parè prove intercategories organiza in a triple category, but 3-cells there are suitable 'commuting cubes', i.e. there is no data. So can you recover some kind of modifications in this setting?
Christian Williams (Sep 12 2022 at 14:42):Oh, I was talking about Dbl a different way... I meant the tricategory of lax functors, lax transformations, and modifications.
Double categories actually form a triple category; this is what I have been working on. Hopefully it is ready to post here in the next few days.
Bruno Gavranović (Sep 14 2022 at 05:48):Thanks @Christian Williams , this might be exactly what I was looking for! I still need to unpack the full details of , but if I understand it correctly - thinking "1-dimensionally" when pondering higher category theory (sounds a bit funny when I say it like this) is the wrong way to think about it.
What double categories seem to give us is an extra level of fidelity compared to 2-categories by allowing us to separately track strict vs lax morphisms. If I understand correctly, this is how they're mainly used, right?
Christian Williams (Sep 14 2022 at 15:25):yes, that's right
Jonas Frey (Sep 14 2022 at 17:25):There's also a 2-category enriched category (a sort of tricategory) of
There's a well behaved notion of biadjunction in this tricategory.
This is summarized eg in Section 2 of my paper on triposes and q-toposes. It goes back to Verity's thesis and can be understood as a special case of the double categorical account discussed above, but I found it easier to handle.