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Stream: learning: questions

Topic: name for globular sets without globular set condition

Avi Craimer (Jun 10 2022 at 19:29):

I'm learning about n-categories and how globular sets play a role analogous to what graphs play in relation to categories. Globular sets are defined as functors into set from something with this shape that also satisfy the globular set conditions. :

polygraph-globular-set-diagram-1.png

source

My question is this. Is there a standard name for a functor from the diagram above into Set without the globular set conditions? I ask because this seems like a natural generalization of a graph with higher order edges and it is surprising to me that there doesn't seem to be a name for it. It is difficult to talk about it as "the category of globular sets without the globular set condition". Intuitively, I would call such structures higher-order graphs or higher-order networks, but these terms are not standard.

I've looked into the terms polygraph and computad because these seem similar, but I still am not sure if either of these is a correct term. Polygraphs (at least in some formulations) appear to include the globular set condition. I can't entirely understand the definition of computads on nlab, but it seems like they are related.

[Update]
After the exchange with @Nathanael Arkor below, I've confirmed that what I'm looking for is not a polygraph/computad, since these include a globularity condition.

Nathanael Arkor (Jun 10 2022 at 19:37):

These are called $n$-graphs (or $\infty$-graphs) by Burroni in the paper introducing polygraphs.

Nathanael Arkor (Jun 10 2022 at 19:38):

Polygraphs generalise higher graphs by permitting $(n + 1)$-cells between chains of $n$-cells. Computads are another term for polygraph.

Avi Craimer (Jun 10 2022 at 19:40):

Thanks @Nathanael Arkor
In that paper on page 44, Burroni says that an ∞-graph has to meet some extra equations/conditions:
image.png

Are these conditions not applied to n-graphs of finite n? It looks to me like the conditions are applied to n-graphs. Since on page 45, his definition of a 2-graph says, "starting from an ∞-graph", so the conditions on the ∞-graph would be enforced for the 2-graph.

Nathanael Arkor (Jun 10 2022 at 19:51):

Yes, they are, but I imagine you want this condition to hold. It says that if you have a 2-cell $\alpha : f \Rightarrow g$, then $f$ and $g$ must have the same domain and codomain as one another.

Nathanael Arkor (Jun 10 2022 at 19:51):

I.e. your cells are globular.

Nathanael Arkor (Jun 10 2022 at 19:52):

Without this condition, you would allow 2-cells like:
image.png

Avi Craimer (Jun 10 2022 at 19:53):

That's right. My original question was about whether there is an official term for higher-order graphs that allow both globular and non-globular structures.

Nathanael Arkor (Jun 10 2022 at 19:55):

Oh, I see, sorry, I misread. I thought you just wanted to avoid the extra generality of polygraphs. I'm afraid I don't know of such a term, in that case.

Avi Craimer (Jun 10 2022 at 19:55):

That's okay. It's a good illustration of how hard it is to talk about without having a term for it!

Avi Craimer (Jun 10 2022 at 19:57):

Your answer was still helpful because it means I can definitely rule out the terms polygraph and computad from my search.

Avi Craimer (Jun 10 2022 at 22:13):

If it turns out there is no existing term for what I describe above, I would also be interested if anybody has opinions on what would be a good term to coin.

Here are some options:

• pre-globular set (suggested by @fosco)
• non-globular n-graph (however, we want to include globular as well)
• higher-order graph

Nathanael Arkor (Jun 10 2022 at 22:28):

"Pre-globular set" is not suggestive of the structure you are describing, because it has nothing to do with globes at all. You could just as well call it "pre-simplicial set", etc.

fosco (Jun 12 2022 at 11:38):

well, it's like a globular set -same basic shapes- but without the globular identities...

fosco (Jun 12 2022 at 11:40):

A "pre-simplicial" set would be "like a simplicial set, same number of maps in each degree, but no simplicial identities", but certainly not just maps $s,t$.

Nathanael Arkor (Jun 12 2022 at 14:27):

fosco said:

well, it's like a globular set -same basic shapes- but without the globular identities...

"Globular" specifically refers to the shapes satisfying the globular identities. This shape isn't very globe-shaped, for instance:
image.png

Nathaniel Virgo (Jun 13 2022 at 00:17):

I don't know anything about globular sets, but I like "higher-order graph" for what you describe, it seems a very natural name.

James Deikun (Jun 14 2022 at 01:22):

They actually end up looking a lot like cubical sets with a bunch of faces missing. I think they end up being in the same family of things as cubical sets only they're "cubical sets without sides", I guess. The same sort of considerations that make people want to add diagonals, connections, etc to cubical sets would make you want to add the sides back into these, I think.

fosco (Jun 14 2022 at 07:46):

Nathanael Arkor said:

fosco said:

well, it's like a globular set -same basic shapes- but without the globular identities...

"Globular" specifically refers to the shapes satisfying the globular identities. This shape isn't very globe-shaped, for instance:
image.png

I see your point now, thanks. I agree with you

Chad Nester (Jun 14 2022 at 08:39):

... "tubular"?