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David Egolf (Jan 03 2021 at 02:05):I'm looking for a resource that introduces double categories, ideally on the level of "Seven Sketches in Compositionality". I'm trying to understand the concept from the nlab, but it's slow going. Something with simpler examples would be especially helpful.
John Baez (Jan 03 2021 at 03:10):Consider taking Simona Paoli's course on double categories at the ACT2021 Adjoint School!
David Egolf (Jan 03 2021 at 04:44):Thanks for the suggestion. Unfortunately, I've got enough on my plate for the moment.
However, I did see that the readings for that course were listed, which may be useful resources:
Structured cospans
A double categorical model of weak 2-categories
Segal-type models of higher categories
This part of the description of the course is appealing: "An approach to bicategories and double categories is available in higher category theory through models that do not require a direct checking of the coherence axioms, such as the Segal-type models." That sounds like it could save a lot of tedious work.
Nathaniel Virgo (Jan 03 2021 at 06:56):I found David Jaz Myers' String Diagrams for Double Categories and Equipments very helpful. It should probably be read in conjunction with other resources, since it's not designed as an introduction to the subject as such, but it's easy to follow and the graphical language makes double categories very intuitive.
John Baez (Jan 03 2021 at 21:51):David Egolf said:
This part of the description of the course is appealing: "An approach to bicategories and double categories is available in higher category theory through models that do not require a direct checking of the coherence axioms, such as the Segal-type models." That sounds like it could save a lot of tedious work.
As usual in category theory, it avoids tedious work by introducing another layer of abstraction. So you have to suffer through learning a bunch more concepts, and then in the end it's supposed to pay off.... and it probably does.
John Baez (Jan 03 2021 at 22:51):I have avoided using Segal-type models in my applications of double categories to concrete systems like electrical circuits, Markov processes, Petri nets etc., because I think practitioners of those subjects would find Segal categories even more off-putting than double categories!
In particular, the simplices in Segal categories are less intuitive than the squares in double categories, where each part - the objects, the vertical arrows, the horizontal arrows and the square itself - has a clear meaning in the applications.
However it might be good to use Segal-type methods to prove some of the theorems I'm having to prove! Some of the proofs are indeed tedious. One could use fancier methods in the proofs, while black-boxing them so practitioners would not need to encounter them.
John Baez (Jan 03 2021 at 23:14):Of course in the end we want to develop a "food chain" so that electrical engineers can talk to applied category theorists who talk to category theorists who talk to homotopy theorists and useful ideas get passed back and forth without the engineers needing to understand Segal categories or the homotopy theorists needing to understand RLC circuits!