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Stream: theory: category theory

Topic: strings for triple categories

Christian Williams (Sep 18 2022 at 18:49):

Hello all, I think I've found a way to draw string diagrams in 3 dimensions. Here is a cube of a triple category. cube.png

Christian Williams (Sep 18 2022 at 18:50):

The cube connects the inner square to the outer square. The four beads are the side faces of the cube.

Christian Williams (Sep 18 2022 at 18:50):

Cubes compose in three ways: horizontal, vertical, and transversal. cube-comp-h.png cube-comp-v.png cube-comp-t.png

Christian Williams (Sep 18 2022 at 18:56):

Let me know what you think!

Christian Williams (Sep 18 2022 at 18:58):

If the cube isn't obvious, here's a picture: cube-exp.png

Mike Shulman (Sep 18 2022 at 21:37):

Interesting. Am I right that the horizontal and vertical compositions don't graphically enforce that the entire faces being composed along match up?

Christian Williams (Sep 18 2022 at 22:14):

It is enforced: you compose along one of the four beads, which is an entire face of the cube. If two cubes are composable along a face, they share a bead on the left & right or top & bottom - but the two will appear as mirror images of each other, because the cube is viewed from inner to outer.

Christian Williams (Sep 18 2022 at 22:18):

cube-comp-h-exp.png

Christian Williams (Sep 18 2022 at 22:26):

You have to learn to read from inner to outer. It feels weird at first, because diagrams appear "upside down" on the top face and "backwards" on the left face; but it's actually been easy to become normal.

Mike Shulman (Sep 18 2022 at 23:09):

What I mean is that it's possible to draw a picture in which those two brown beads are different. The fact that they have to be the same is an extra requirement, not imposed by the shape of the pictures.

Mike Shulman (Sep 18 2022 at 23:10):

In contrast to how in an ordinary string diagram, the fact that domains and codomains match in a composition is enforced by the fact that each string in the picture has a single label, rather than an extra requirement that two distinct objects in the picture must have the same label.

Christian Williams (Sep 18 2022 at 23:24):

I understand.

Okay, so I drew it this way to show the face of composition; but in practice, "the middle disappears" when you compose - these two beads matching is a rule on composition, and then you just get a cube like this: cube-comp-h-red.png

Mike Shulman (Sep 19 2022 at 03:56):

Does that version provide a place to label the vertical arrow in the "outside" of the face being composed along?

Matteo Capucci (he/him) (Sep 19 2022 at 09:50):

Christian Williams said:

Hello all, I think I've found a way to draw string diagrams in 3 dimensions. Here is a cube of a triple category. cube.png

ha, very clever!

Christian Williams (Sep 19 2022 at 15:58):

Mike Shulman said:

Does that version provide a place to label the vertical arrow in the "outside" of the face being composed along?

No - I first drew the "rule of composition", conditioned by matching in the middle, as in $x\to y\to z$. But then in this cube above it disappears, as in $x\to z$; it can no longer be composed along, so it is not needed in the diagram.

But just as in ordinary composition, one can still use supplemental notation to record how the composite was formed, ie $\Gamma \ast_\tau \Delta$ for horizontally composing along the transformation $\tau$.

Christian Williams (Sep 19 2022 at 16:05):

so you have two ways of drawing, one "reduced" with no conditions and you lose the middle label, and one "full" with the middle label plus the condition that the two face-beads match. I think this is all analogous to lower dimensions.

Mike Shulman (Sep 19 2022 at 16:06):

In an ordinary string diagram, the string being composed along is still present and labeled in the diagram.

Mike Shulman (Sep 19 2022 at 16:06):

I'm not trying to be too critical of your syntax; it looks intriguing! I'm just trying to understand the differences.

Christian Williams (Sep 19 2022 at 16:10):

yes, I understand. I think there's just a couple necessary but small caveats to drawing in the third dimension.

Christian Williams (Sep 19 2022 at 16:11):

one just has to understand that composing is matching faces, not only edges.

Christian Williams (Sep 21 2022 at 00:04):

update: the general form of a cube is a bit more than I first drew; see below.
cube-2.png

Christian Williams (Sep 21 2022 at 00:06):

the outward edges and faces can be arbitrary composites, so we need a notation to delineate where the face starts and stops. for now, using dotted black lines seems to work fine.

Christian Williams (Sep 21 2022 at 00:08):

for example, here is the associator of a double category, an invertible 3-cell: cube-assoc.png

Christian Williams (Sep 21 2022 at 00:10):

I think it also helps to make the third dimension more clear.

Christian Williams (Sep 22 2022 at 23:40):

yesterday @Mike Shulman noticed that so far these cubes have composite domain 2-cell but only a single codomain 2-cell (the outer square, with a big transparent "bead" covering up the inner square). how can we draw a cube with a composite codomain?

Christian Williams (Sep 22 2022 at 23:43):

well one way to get a cube from a composite to a composite, we can of course compose two cubes, as drawn above horizontally and vertically. then, how do we generalize that picture to denote that the connecting faces are not just composites of single ones for each part of the composite?

Christian Williams (Sep 22 2022 at 23:44):

here's my attempt so far - just draw one big bead from the inner composite to the outer composite. comp-cod.png

Christian Williams (Sep 22 2022 at 23:44):

it may look silly, but it might be sufficient. I'm not sure

Christian Williams (Sep 22 2022 at 23:53):

I think there's probably no need for the bending, actually; just a big bead straight across

Christian Williams (Sep 23 2022 at 00:07):

okay, I was misleading myself by starting with a horizontal composite. there is no need for the middle brown bead, it's much simpler.

Christian Williams (Sep 23 2022 at 00:09):

hm, but you still need the brown string to show that the outer face is a composite.

Christian Williams (Sep 23 2022 at 00:16):

comp-cod-2.png

Christian Williams (Sep 23 2022 at 00:19):

the middle dotted lines mean "no connection between light green and dark green, except for however they are connected by the big bead"

Christian Williams (Sep 23 2022 at 00:35):

it's like a bridge passing over, and the light colors passing under, no connection between them

Christian Williams (Sep 23 2022 at 00:37):

whatever notation works best so that the brown string can be there, to show the outer face as a composite

Mike Shulman (Sep 25 2022 at 00:22):

I don't understand what this means.

Christian Williams (Sep 25 2022 at 16:29):

to draw a cube whose source and targets are both horizontal composites, I'm saying all we need to add to the picture is the vertical morphism along which the target square is a composite. and I'm noticing that it's consistent to simply draw the brown string there between them, floating above the source square; it's not blocking the view of anything below it, because displaying the composite source square requires duplicating its middle vertical morphism (the beige string connecting light green to light yellow) inside each "window bead" of the target.

Mike Shulman (Sep 26 2022 at 04:21):

Could you perhaps give names to all the objects and morphisms and cells? I have a lot of trouble referring to them as "the brown string" and "the beige string".

Mike Shulman (Sep 26 2022 at 04:21):

And label them in the diagram?

Christian Williams (Sep 26 2022 at 16:53):

ah yes, sorry about that. here it is cube-hcomp-label.png

Christian Williams (Sep 26 2022 at 16:54):

the source is the composite $HV_{11} \circ HV_{21}$, and the target is the composite $HV_{12}\circ HV_{22}$

Christian Williams (Sep 26 2022 at 16:57):

the cube is framed by $HT_1: H_{11}\circ H_{31}\Rightarrow (H_{12}\circ H_{32})(T_1,T_3)$, etc. (I forgot to label the transversal $T_1, ..., T_4$ but one can infer them.

Christian Williams (Sep 26 2022 at 17:03):

any way, what I was saying is that if we duplicate $V_2$, so the beads of the source composite sit inside the beads of the target composite, then we can include $V_4$ in the drawing, the string along which the target beads are composed. we only need some notation like these dotted lines to indicate "no connection between $V_2$ and $V_4$, former is in the source and latter is in the target"

Christian Williams (Sep 26 2022 at 17:05):

it's just like poking two holes in a box, rather than one.

Mike Shulman (Sep 26 2022 at 23:57):

Christian Williams said:

the source is the composite $HV_{11} \circ HV_{21}$, and the target is the composite $HV_{12}\circ HV_{22}$

I don't see anything labeled $HV_{11}$ or any of those other things.

Could you maybe draw the same thing in ordinary notation, so I can compare?

Christian Williams (Sep 27 2022 at 00:21):

cube-hcomp-diagram.png

Mike Shulman (Sep 27 2022 at 15:27):

Thanks! So where are the cells $HV_{11}, HV_{12}, HV_{21}, HV_{22}$ notated in the string diagram picture?

Christian Williams (Sep 27 2022 at 17:38):

$HV_{11}$ and $HV_{21}$ are the two inner beads, $HV_{12}$ and $HV_{22}$ are the two outer beads.

Mike Shulman (Sep 27 2022 at 17:40):

Can you label them please?

Christian Williams (Sep 27 2022 at 18:14):

cube-hcomp-labels.png

Mike Shulman (Sep 27 2022 at 19:33):

Okay. So what would you draw if the transversal domain is just a single cell and the codomain is a composite pair of them?

Christian Williams (Sep 27 2022 at 19:35):

we can make either one the inner beads the identity. the two choices are equal

Mike Shulman (Sep 27 2022 at 19:36):

What if the transversal domain is a composite of three cells?

Mike Shulman (Sep 27 2022 at 19:37):

Or if the transversal domain is the composite of two cells horizontally and the transversal codomain is the composite of two cells vertically?

Christian Williams (Sep 27 2022 at 19:46):

I'm proposing that in general we can add identities as needed, to express domain and codomain as the same form of composite

Mike Shulman (Sep 27 2022 at 19:47):

That seems very artificial.

Mike Shulman (Sep 27 2022 at 21:52):

I think I would find it easier to read with the codomain just given as a separate 2D diagram if necessary.

Christian Williams (Sep 27 2022 at 22:18):

I really think it's fine. identities are very useful in 2D strings, to make more clear standard forms. std-form.png

Christian Williams (Sep 27 2022 at 22:21):

in all my drawings, beads are rectangles, giving both domain and codomain the same amount of space, even if they are very different sizes of composites. it gives each concept a clear and standard appearance.

Christian Williams (Sep 27 2022 at 22:21):

now the issue is just more obvious in 3D, because there is the aspect of depth and visibility. but it's the same idea

Christian Williams (Sep 27 2022 at 22:27):

if you see a serious problem, let me know. otherwise it's a matter of preference; one can always slice the cube as needed.

Mike Shulman (Sep 28 2022 at 01:27):

Indeed, I was just letting you know what my preference would be. (-:

Morgan Rogers (he/him) (Sep 28 2022 at 08:38):

I'm impressed at how much you've managed to expressed in a (beautifully colourful) planar diagram @Christian Williams! How are you producing these?

Christian Williams (Sep 28 2022 at 14:23):

thanks! I'm drawing them on Notability

Christian Williams (Sep 28 2022 at 14:55):

when I finish the thesis, I hope to work with a programmer to make an interactive app for string diagrams