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Daniel Plácido (Jun 09 2021 at 17:39):Anybody got interesting examples of 2-functors, or even better, lax functors between bicategories? I'm writing my master thesis in these topics but most examples are generally universally derived from other places; are there examples inherently bicategorical, shedding brightness to the importance of the data? Thanks.
Daniel Plácido (Jun 09 2021 at 17:51):What I have so far: functors, lax monoidal functors, strict 2-functors, monads/comonads, lax functors between span/cospan bicategories, Grothendieck fibrations, Duskin nerve, and some others in Bénabou's text
John Baez (Jun 09 2021 at 18:21):@Jade Master and I showed that black-boxing of open Petri nets is lax functor between bicategories.
John Baez (Jun 09 2021 at 18:22):Actually we showed it's a lax functor between pseudo double categories but you can extract a bicategory from any pseudo double category and then a lax functor between pseudo double categories gives a lax functor between their bicategories.
John Baez (Jun 09 2021 at 18:24):There are also examples of 2-functors between bicategories in the "examples and applications" section of Structured versus decorated cospan categories. Most of these examples come from subjects like chemistry, electrical engineering and epidemiology.
Daniel Plácido (Jun 10 2021 at 14:13):Thanks John. I actually went over Structured vs. decorated in some depths due to the Adjoint School, the examples are indeed interesting.
Daniel Plácido (Jun 10 2021 at 14:16):I'm also finding fun examples at Higher-Dimensional Algebra V: 2-Groups, specially the automorphism 2-group of an object in a bicategory. The examples from Chern-Simons seem interesting and remind me to look at higher gauge theory stuffs
Daniel Plácido (Jun 10 2021 at 14:34):related https://math.stackexchange.com/questions/148134/concrete-examples-of-2-categories
Martti Karvonen (Jun 10 2021 at 15:05):Daniel Plácido said:
I'm also finding fun examples at Higher-Dimensional Algebra V: 2-Groups, specially the automorphism 2-group of an object in a bicategory. The examples from Chern-Simons seem interesting and remind me to look at higher gauge theory stuffs
Taking the automorphism 2-group of an object doesn't in general define a 2-functor out of the bicategory for the same reason as taking the automorphism group doesn't define a functor out of a category -- there is no obvious way of defining the action on morphisms. Isotropy groups provide a fix, and the same story can be generalized to the 2-d setting. Taking automorphism groups of objects is functorial for a groupoid (and coincides with isotropy in this case), so taking automorphism 2-groups should also work for a 2-groupoid (and should coincide with 2-d isotropy).
Simon Willerton (Jun 10 2021 at 16:05):I don't know if this is the kind of thing you want, but how about the representation functor from some version of Bimod to CAT?
In a bit more detail, consider the bicategory with finite groups, say, as objects, finite dimensional bimodules (over the complex numbers, say) as one-morphisms, and bimodule maps as two-morphisms. You can construct a 2-functor to the 2-category of categories, where a group goes to, say, its category of finite dimensional complex representations, a bimodule goes to the corresponding functor and a bimodule map goes to the corresponding natural transformation.
(I'm using the words 'representation' and 'module' interchangeably - so I'm not trying to convey any subtle distinction!)
Tom Hirschowitz (Jun 14 2021 at 07:11):I think Glynn Winskel has one in this paper, from games and strategies to cats and profunctors.
Peiyuan Zhu (Aug 29 2021 at 15:14):Which book would you recommend to study gauge theory?
John Baez (Aug 30 2021 at 02:18):I like the book I wrote with Javier Muniain: Gauge Fields, Knots and Gravity.