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Stream: theory: category theory

Topic: What do you call an object of Joyal's category $$\Theta$$?

Tim Campion (Feb 22 2023 at 17:57):

We have:

• An object of the simplex category $\Delta$ is called a simplex.
• An object of a cube category $\square$ is called a cube.
• An object of the globe category $\mathbb G$ is called a globe (or sometimes a cell).
• An object of Joyal's category $\Theta$ is called what?

Whatever the answer I want it to be applicable to presheaves too as follows (where either $\mathcal C = \mathsf{Set}$ or else I'm talking about generalized elements):

• An element $x \in X([n])$ is a simplex in $X$, where $X : \Delta^{op} \to \mathcal C$ is a simplicial object and $[n] \in \Delta$.
• An element $x \in X(\square^n)$ is a cube in $X$, where $X : \square^{op} \to \mathcal C$ is a cubical object and $\square^n \in \square$.
• An element $x \in X(\mathbb G_n)$ is a globe in $X$, where $X : \mathbb G^{op} \to \mathcal C$ is a globular object and $\mathbb G_n \in \mathbb G$.
• An element $x \in X(\theta)$ is a what? in $X$, where $X : \Theta^{op} \to \mathcal C$ is a $\Theta$-presheaf and $\theta \in \Theta$.

I suppose that presheaves on $\Theta$ are often called "cellular sets", and so maybe the answer should be "cell". But this is a bit suboptimal because it clashes with the use of "cell" to mean "globe", and also with the more generic use of "cell" in the setting of an arbitrary presheaf category.

$\Theta^{op}$ is a category of disk bundles, but even ignoring the contravariance, I'm not sure "disk bundle" would work in the second set of bullet points. Probably whatever the term, it should be one word.

Amar Hadzihasanovic (Feb 22 2023 at 19:34):

Joyal called them “Batanin cells”.
Personally I would call them “diagrams of globes”.

Amar Hadzihasanovic (Feb 22 2023 at 19:37):

The convention I've been using is “(simplicial/cubical/globular) cell in $X$” for your simplex/cube/globe in $X$, and “diagram in $X$” for a pasting diagram containing multiple (maximal) cells, which the $\theta$ are.

Amar Hadzihasanovic (Feb 22 2023 at 19:38):

I'd be happy for that to catch on.

Jonas Frey (Feb 22 2023 at 21:06):

I've seen them called "globular pasting schemes", e.g. here

Mike Shulman (Feb 22 2023 at 21:30):

If you wanted to try to coin a word, it could be something like "globuplex", since it contains both globes and simplices....

David Michael Roberts (Feb 22 2023 at 23:52):

@Mike Shulman what would the plural be?? Globuplices?

Mike Shulman (Feb 22 2023 at 23:59):

That's pretty awful, isn't it? You could probably say "globuplexes" which is marginally less terrible.

John Baez (Feb 23 2023 at 07:00):

I vaguely remember seeing some word for this kind of shape that's a bit less unpleasant than "globuplex".

Matteo Capucci (he/him) (Feb 23 2023 at 08:52):

Globplex? Rolls off the tongue a bit better

Morgan Rogers (he/him) (Feb 23 2023 at 09:04):

Or if you would like to reverse the order of combining words, "simplobe" has an amusing ring to it.

Amar Hadzihasanovic (Feb 23 2023 at 10:31):

I find it confusing to think of $\Theta$ as "containing simplices", at least as long as one thinks of its objects as pasting diagrams.
Unlike $\Delta$, $\square$ and $\mathbb{G}$ which are generated by "(co)face" and "(co)degeneracy" morphisms, $\Theta$ also has "(co)composition" morphisms which are not (co)faces of any meaningful diagram.
The inclusion of $\mathbb{G}$ into $\Theta$ sends faces to faces and degeneracies to degeneracies, but the full and faithful functor from $\Delta$ to $\Theta$ only sends the "faces with consecutive vertices" to faces, so it changes the geometric picture somehow.
I think it's better to think of it as a representation of the simplex category, rather than an inclusion of the simplices into $\Theta$.

Amar Hadzihasanovic (Feb 23 2023 at 10:34):

In fact this representation is essentially the same as the full and faithful functor from $\Delta$ into $\mathbf{Cat}$ (or $\mathbf{Pos}$) sending the $n$-simplex to the linear order on $(n+1)$ elements, and I don't think it's helpful to think of $\mathbf{Cat}$ or $\mathbf{Pos}$ as “containing the simplices”.

Especially considering that $\omega\mathbf{Cat}$ “contains the simplices” at least in a couple of other ways (i.e. Street's oriented simplices and their duals).

Amar Hadzihasanovic (Feb 23 2023 at 10:39):

In any case, I think the objects of $\Theta$ have already been called

• Batanin cells,
• Batanin trees,
• globular pasting schemes,
• simple $\omega$-categories,
• simple $\omega$-graphs,
• disks (or Joyal disks),

so any additional “unique” name will probably only contribute in the way described by this xkcd strip.

Morgan Rogers (he/him) (Feb 23 2023 at 11:11):

...I still like "simplobes"...

Tim Campion (Feb 23 2023 at 14:39):

multiglobe?

Tim Campion (Feb 23 2023 at 14:50):

grid?

Tim Campion (Feb 23 2023 at 14:52):

Maybe an object of $\Theta$ is just a "theta"? I dunno... is "Let $\theta \in \Theta$ be a theta...." okay? Or "Let $x \in X(\theta)$ be a theta in $X$...."?

Amar Hadzihasanovic (Feb 23 2023 at 16:02):

In rewalt I did just call their class Theta...

c9eb93bf-b2bd-482b-8481-cedc723359b2.jpg

Fernando Yamauti (Feb 23 2023 at 18:28):

In the context of test categories, I've heard before "polyglobe" being used for the free monoidal category generated by globes with the point as the unity.

Matteo Capucci (he/him) (Feb 23 2023 at 19:31):

Amar Hadzihasanovic said:

In any case, I think the objects of $\Theta$ have already been called

• Batanin cells,
• Batanin trees,
• globular pasting schemes,
• simple $\omega$-categories,
• simple $\omega$-graphs,
• disks (or Joyal disks),

so any additional “unique” name will probably only contribute in the way described by this xkcd strip.

If I can vote, 'Joyal disks' sounds the nicest to me

Matteo Capucci (he/him) (Feb 23 2023 at 19:31):

After 'simplobes', of course

Mike Shulman (Feb 23 2023 at 20:37):

I think if you think of a 2-simplex as two composable arrows (together with their composite), then it makes perfect sense to think of $\Theta$ as containing simplices.

Jonas Frey (Jun 17 2023 at 04:02):

Matteo Capucci (he/him) said:

Amar Hadzihasanovic said:

In any case, I think the objects of $\Theta$ have already been called

• Batanin cells,
• Batanin trees,
• globular pasting schemes,
• simple $\omega$-categories,
• simple $\omega$-graphs,
• disks (or Joyal disks),

so any additional “unique” name will probably only contribute in the way described by this xkcd strip.

If I can vote, 'Joyal disks' sounds the nicest to me

I think the topos of cellular sets is the classifying topos of a class of structures that Joyal calls disks, but the category $\Theta$ is actually dual to the category of finite disks, in the same way as $\widehat\Delta$ classifies finite strict bounded linear orders, but $\Delta$ is opposite to the category of finite strict bounded linear orders.