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Jacob Zelko (Mar 27 2024 at 16:48):Hi folks!
@Evan Patterson and I were speaking recently about double categories and we have been trying to think of resources for building a better foundation in Double Categories. Especially for folks who have built an understanding in the basics of category theory (e.g. functors, universal properties, pullbacks/pushouts, equalizers, duality, etc.) and want to continue into higher category theory. One resource we thought of was:
Any other resources folks can think of here? Cheers! :deciduous_tree:
Joshua Meyers (Mar 27 2024 at 16:59):I am in this exact boat right now and I found the paper "Pseudo-categories" by Nelson Martins Ferreira very helpful, in order to see in full rigor how weak double categories, double functors, and natural transformations work
Joshua Meyers (Mar 27 2024 at 17:00):Also Mike Shulman's paper "Framed Bicategories and Monoidal Fibrations" of course
Jacob Zelko (Mar 27 2024 at 17:06):Amazing! Thank you so much @Joshua Meyers! Are you still actively reading through these texts? Or have you finished them already?
Joshua Meyers (Mar 27 2024 at 17:28):I am neither actively reading them nor have I finished them already. I read some of them and now use them as reference.
Peva Blanchard (Mar 27 2024 at 17:28):I just started learning about double categories (like a week ago). Building a list of learning resources would be very helpful.
I'm reading "Framed bicategories and monoidal fibrations" by Mike Shulman.
@Evan Patterson 's blog post and talk on double categorical databases has been also very helpful to ground these higher things to "concrete" things.
Joshua Meyers (Mar 27 2024 at 17:29):I also want to agree that Higher-Dimensional Categories by Marco Grandis is a great resource
Alex Kavvos (Mar 28 2024 at 00:57):I want to third the suggestion for Mike Shulman's paper on framed bicategories
Evan Patterson (Apr 03 2024 at 00:45):Peva Blanchard said:
Evan Patterson 's blog post and talk on double categorical databases has been also very helpful to ground these higher things to "concrete" things.
Glad it was helpful! On that note, Michael Lambert and I have just finished cleaning up our old draft/blog post writings and put out a short paper about double-categorical databases: https://arxiv.org/abs/2403.19884
One way to read this paper is as a friendly and example-focused introduction to cartesian equipments, which will hopefully be a useful complement to the more conventional math literature on this topic.
Ryan Wisnesky (Apr 03 2024 at 01:29):Hey @Evan Patterson , is the exact expressive power double relational ologs known? i.e., is a statement of the internal language available? As a formalism I'd like to compare it with containments of expressions in multi-sorted binary relation algebra, allegories, etc.
Evan Patterson (Apr 03 2024 at 05:33):Hi Ryan, it's a good question. I don't think a fully precise answer is available yet but the answer must be something like "typed regular logic with equality and function and relation symbols."
In his earlier paper on double categories of relations, Michael characterizes the 'double categories of relations' satisfying a number of extra conditions (have tabulators, are "functionally complete", have a "subobject comprehension scheme") as precisely the double categories of relations on a regular category, namely the regular category of objects and arrows in the double category (Theorem 10.2). And the close connection between regular categories and regular logic is a classic part of categorical logic. But AFAIK these pieces have not been put together in a completely clear way.
Evan Patterson (Apr 03 2024 at 05:42):It's even more interesting to ask what happens when these extra conditions are dropped. Category theoretically, it's just as easy to work in a generic cartesian equipment as in a 'double category of relations.' But a cartesian equipment can accomodate things like spans and even profunctors. The internal language of these is mysterious, though there has been work on a type theory for virtual equipments.