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Nathaniel Virgo (Jun 20 2021 at 02:01):Hi all
I'm wondering if there's a good reference / introduction to the various notions of functor between (weak) double categories. nlab mentions that they come in various flavours (strict, lax, oplax etc.) as for bicategories, which makes sense, but it doesn't get very far in terms of defining them.
Mostly what I'm looking for is just the definitions spelt out explicitly enough that I (being a person without much sophistication) can work my way through them step-by-step in the context of some example double categories that I'm familiar with and get a feel for what they're like.
I'm also very interested to know what the corresponding notion(s) of natural transformations are for the various notions of functor between double categories, so if there is something that goes into detail about that as wel it would be great.
Does a resource like that exist?
Mike Shulman (Jun 20 2021 at 03:59):My Framed bicategories and monoidal fibrations has more than the nLab, at least, plus a number of examples. Another good reference is Grandis & Pare Adjoints for double categories, which assembles lax and oplax double functors into a double category of double categories.
Nathaniel Virgo (Jun 20 2021 at 08:59):Also as a self-answer, I found a nicely explicit treatment in the book "Higher Dimensional Categories: From Double to Multiple Categories" by Marco Grandis. (Weak double categories are defined in definition 3.3.1 and the various notions of functor in definition 3.5.1)
Daniel Plácido (Jun 21 2021 at 15:36):The appendix to Structured cospans has nice, spelled out basic double categorical definitions.
John Baez (Jun 21 2021 at 16:33):Thanks! Even better is the appendix to Kenny Courser's thesis Open Systems: A Double Categorical Perspective. It's not mainly about different kinds of double functors, but it has a lot of basic definitions in double category theory.
Matt Earnshaw (Jun 21 2021 at 17:32):for a different flavour, @Ed Morehouse has a nice presentation using string diagrams in this preprint
Nathaniel Virgo (Jun 22 2021 at 03:07):Oh, great, that Ed Morehouse paper is going to save me a lot of time working out the string diagrams for myself!
Graham Manuell (Jun 22 2021 at 13:16):All of these resources seem to assume that the double functors are strict in the one direction. Is there a reason for this? I would have naively assumed it would be better for compositions in both directions to be preserved only up to isomorphism.
Nathaniel Virgo (Jun 22 2021 at 13:42):I guess because double categories themselves are strict in one direction, so you sort of have to be strict along that direction. nlab mentions a few ways to change that, though.
Graham Manuell (Jun 22 2021 at 13:49):Thanks, though I don't know why that would mean you have to be strict along that direction. When dealing strict 2-categories it is still usual to consider pseudofunctors instead of only strict 2-functors.
Nathaniel Virgo (Jun 22 2021 at 14:22):I might have spoken too soon. (I am still learning this stuff.) That makes sense. Now I have the same question and I hope someone with more expertise can answer it.
Mike Shulman (Jun 22 2021 at 15:45):It's possible to define "double pseudofunctors", e.g. in my Comparing composites of left and right derived functors, or in Verity's Enriched categories, internal categories and change of base (where the double categories are weak in both directions as well). I think double pseudofunctors aren't as common in the literature because, perhaps surprisingly, they don't come up in practice all that often. Also, double categories are the algebras for a 2-monad on , and the pseudo/lax/colax morphisms for that 2-monad are the double functors that are strict in one direction, so the 2-categories of double categories and those morphisms have good formal properties.