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Stream: theory: category theory

Topic: Higher string diagrams

Simon Burton (Mar 07 2026 at 16:49):

What I'm wondering about here splits into two parts:

(1) Is the syntax of 3d string diagrams sound for reasoning in strict monoidal 2-categories (or strict 3-categories) ?

(2) In particular, is the cartesian 2-category of categories one of these strict monoidal 2-categories ? By "is" here i might mean up to some obvious cartesian strictification.

For the first part, i think it's reasonable to require string diagrams to come with enough "grid-like" structure so that we are not obsessing over Morse theory of stratified manifolds etc. So these string diagrams are essentially already combinatorial, and anyone who wants to insert Alexander's horned sphere in their string diagram is doing something different.

David Corfield (Mar 07 2026 at 18:09):

Perhaps of interest, there's the approach of Christoph Dorn et al. as described at An Invitation to Geometric Higher Categories.

Simon Burton (Mar 08 2026 at 10:48):

Yes, very interesting, thankyou. I am generally doing a lot of wandering around in the literature and getting lost in the process. Maybe what I need to do is focus on the datastructure(s) underlying strict 3-categories: globular sets, computads, pasting diagrams, (??) ... Then whatever that is, I can start to see how close this is to higher string diagrams as a datastructure...

Simon Burton (Mar 08 2026 at 11:01):

This is another nice reference "Surface diagrams for gray-categories" by Hummon. Quoting from Chapter 4: "Surface diagrams are a tool used to describe composites of 3-morphisms between composites of 2-morphisms. We begin by interpreting surface diagrams in the strict world of 3-categories. It’s relatively easy to write down a theory of such diagrams." So this seems promising...

John Baez (Mar 08 2026 at 21:10):

You should also check out Todd Trimble's articles on surface diagrams on his section of the nLab.

Simon Burton (Mar 10 2026 at 10:28):

Perhaps i should have put this in the topic title, but i'm really only considering string diagrams for strict (higher) categories... As Hummon writes, on page 30, in a 3-category the following two expressions are the same:
image.png

Simon Burton (Mar 10 2026 at 10:32):

this is because we don't have enough identity maps to anchor the ends of the braids. The identity maps are just equalities which can be freely inserted/deleted and so these braids just unwind. I'm not sure if this is more or less confusing than the weak case!

John Baez (Mar 10 2026 at 23:04):

Wow, so you really are studying string diagrams for strict monoidal 2-categories!?! You indeed said that, but I didn't notice because it's so wild. Not only do we have the double braiding equal to the identity, as shown in your cube diagram, even the single braiding is equal to the identity. That is, for any pair of objects

$x \otimes y = y \otimes x$

and the braiding isomorphism

$B_{x,y} : x \otimes y \to y \otimes x$

is equal to the identity! If I had the energy I could draw a cube picture of this....