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Stream: theory: category theory

Topic: I need a name for the following

Tim Campion (Jul 11 2022 at 16:59):

Let $A$ be a strict $n$-category, and let $B = A \amalg_{\partial \mathbb G_n} \mathbb G_n$ be the result of gluing a copy of the $n$-globe $\mathbb G_n$ onto $A$ via some "attaching map" $\phi : \partial \mathbb G_n \to A$. Let $\nu$ denote the "new" $n$-globe of $B$ (i.e. the one we freely glued on). For $0 \leq k \leq n-1$, let $\circ_k$ denote $k$-composition in $B$ -- i.e. it's the composition operation which requires the appropriate $k$-dimensional boundaries to agree.

I'm interested in studying certain "presentations" of $n$-morphisms in $B$, which for the moment I'll call "doohickeys", defined inductively as follows. First, $\nu$ is "presented "by a unique doohickey of "order" $-1$. Inductively, if $g$ is presented by a doohickey of "order" $k-1$, then the composite globe $a_+^{(k)} \circ_{k} g \circ_{k} a_-^{(k)}$ is presented by a doohickey of "order" $k$ whenever $a_-^{(n)},a_+^{(k)} \in A_n$ are $n$-cells such that this composite is defined.

So a doohickey is an $n$-morphism of $B$ equipped with a choice of a way to present it in a particular way as a composite of $\nu$ with $n$-morphisms $(a_-^{(0)},a_+^{(0)}, \dots , a_-^{(n-1)},a_+^{(n-1)})$ of $A$. The question is

What should "doohickeys" actually be called? And what should the "order" of a doohickey be called? And is there a better term than "presentation" for what I'm describing above?

Tim Campion (Jul 11 2022 at 17:06):

Some notes:

1. In the cases I'm interested in, every $n$-morphism of $B$ which is not in $A$ is presentable by a doohickey, but this is not true in general. For instance, if $\nu$ is composable with itself, then the composite $\nu \circ_i \nu$ is not presentable by a doohickey.

2. Makkai called something very similar a "pre-atom". The difference is that Makkai studies a situation where one is gluing on many $n$-morphisms at once, and he reserves "pre-atom" for the case where $a_-,a_+$ are less than $n$-dimensional. This is because he will then build up "molecules" from the "atoms" presented by his "pre-atoms". I'm not wild about this terminology because I like my "atoms" to be _indecomposable_, whereas the globes presented by doohickeys are very much decomposable.

Tim Campion (Jul 11 2022 at 17:10):

3. Makkai's terminology has some nice things going for it, though. For instance, you might extend the physics analogy by calling (as Makkai does) $\nu$ the "nucleus" of the atom. You might also think of the morphisms $a_\epsilon^k$ which go into a "doohickey" as being at different "energy levels" for different values of $k$ (I think probably higher $k$ would correspond to a higher energy level than lower $k$).

Tim Campion (Jul 11 2022 at 17:13):

4. I've experimented "pre-cell", which is decent. I've also thought about using "peapod" because that's what my drawing of a doohickey looks like in dimension $n=2$ -- but that analogy doesn't seem to go too far. I don't know if there's some kind of cell biology analogy to be made.

Tim Campion (Jul 11 2022 at 17:15):

5. The thing that I'm doing with doohickeys is to describe (under restrictive hypotheses) the precise equivalence relation on doohickeys which you need to quotient by to get the set of $n$-morphisms of $B$ which are not in $A$. The equivalence relation as I've written it seems pretty suggestive: it's the reflexive symmetric transitive closure of what you get when you allow one of the $a_\epsilon^{(k)}$'s to be "absorbed" (via an appropriate pasting) into $a_\delta^{(k+1)}$ -- somehow like an atom passing to an "excited state" or something.

Tim Campion (Jul 11 2022 at 17:19):

6. I think the main thing which is awkward about adopting Makkai's terminology to this situation is that the place where the "atomic physics analogy" lives (cf. (5) above) is at the level of doohickeys. So I'd want to call a doohickey an "atom" rather than a "pre-atom". But then it feels kind of awkward that "the thing presented by a doohickey/atom" is a "cell/globe/$n$-morphism", none of which are terms which naturally sound like "a thing presented by an atom". (That plus (as mentioned in (2) above) the unfortunate point about "atoms" being decomposable... but I suppose this is unavoidable if one really wants to use an atomic physics analogy.)

Tim Campion (Jul 11 2022 at 17:34):

7. In the atomic orbital analogy, the number $k$ which I called the "order" might be thought of as being like the principal quantum number, which fits well with the idea of "energy levels". It's then a bit unfortunate that the next quantum number -- the azimuth is, in actual physics, a number in a restricted interval rather than $\{-,+\}$-valued. And the quantum numbers which are $\{-,+\}$-valued, like spin, don't really have names which fit very well here.

8. Maybe I could think of $n$ as the "principal quantum number" $n$, the "order" $k$ as the "azimuth" $\ell$, and then regard each cell $a_i^\epsilon$ as being located at an "angle", because the next atomic number $m$ is associated to angular momentum. This almost works out perfectly -- in atomic physics, $m$ is an integer between $-\ell$ and $\ell$. I think this actually works out for me if I shift my indexing by $1$, to think of $\nu$ itself as being associated to "angle number" $m=0$; then forming $a_+^{(k)}\circ_k g \circ_k a_-^{(k)}$ from $g$ is like providing angular momentum data at $m=k+1$, both for $-(k+1)$ and $+(k+1)$...

(8) is a tortured analogy for sure, even though it works hilariously well! Maybe it would be distracting to fit the terminology to it...

Amar Hadzihasanovic (Jul 12 2022 at 08:04):

I've lost you a bit on the last part but I can tell some things:

• I would not suggest going for atom and molecule because that's also the terminology that Steiner used in the 90s and that I recuperated for certain combinatorial encodings of shapes of pasting diagrams. (It is also exactly correct that in this case the "atoms" are indecomposable, so it seems like a better usage).

Amar Hadzihasanovic (Jul 12 2022 at 08:08):

• So if I understand correctly your "doohickeys" are particularly shaped diagram in $B$, where the "outer part" actually has image in $A$. In one preprint, I considered a similar diagram shape, which is different only in that the $a_\alpha^{(k)}$ are supposed to be $k$-dimensional cells; with this provision, if $\nu$ is $n$-dimensional, I called one of your "order $n$ doohickeys" a shell for $\nu$.

Amar Hadzihasanovic (Jul 12 2022 at 08:09):

That preprint got superseded by one in which I did not use the “shell” construction, so as far as I'm concerned you are free to pick up the name if you like it.

Amar Hadzihasanovic (Jul 12 2022 at 08:10):

• The “order” of a doohickey is exactly what Steiner calls the frame dimension of its shape as a pasting diagram.

Amar Hadzihasanovic (Jul 12 2022 at 08:10):

(That includes the fact that the frame dimension of the “atom” $\nu$ is -1!)

Amar Hadzihasanovic (Jul 12 2022 at 08:14):

So my naming suggestion would be to say something like “A shell of order $k$ for $\nu$ in $A$”. You can even still keep your orbital analogies since electron shell is used in physics.

Amar Hadzihasanovic (Jul 12 2022 at 08:15):

(Or “a shell of order $k$ around $\nu$”, if you want.)

Tim Campion (Jul 12 2022 at 15:22):

Amar Hadzihasanovic said:

• So if I understand correctly your "doohickeys" are particularly shaped diagram in $B$, where the "outer part" actually has image in $A$. In one preprint, I considered a similar diagram shape, which is different only in that the $a_\alpha^{(k)}$ are supposed to be $k$-dimensional cells; with this provision, if $\nu$ is $n$-dimensional, I called one of your "order $n$ doohickeys" a shell for $\nu$.

Thanks, I do like the term "shell"!

The dimensionality restriction you have on your "shells" (the condition that $a_\alpha^{(k)}$ be $k$-dimensional) is, I think, the same dimensionality restriction which Makkai puts on on his "pre-atoms" (I presume you're using the same indexing as Makkai, which is off by one from the indexing I've been using) . So I guess the difference is that your "shells" are a bit closer to doohickeys in that you're allowing for the case where $A$ already has some $n$-dimensional cells, which can appear in the top position (as $a_\alpha^{(n)}$ in Makkai's indexing, or $a_\alpha^{(n-1)}$ in my indexing)?

I'm not including this dimensionality restriction in the definition of a doohickey. This doesn't allow me to represent more cells, but rather just gives me a bit more flexibility in how to represent them and describe the equivalence relation on them by which one quotients to get the cells of $B$. I should think a bit about how essential this flexibility is to my approach...

Tim Campion (Jul 12 2022 at 15:26):

@Amar Hadzihasanovic Given a shell $\sigma$ around $\nu$, do you have a name for "the morphism of $A$ appearing as $a^{(k)}_\epsilon$ in $\sigma$"?

Tim Campion (Jul 12 2022 at 15:28):

Or, given $a \in A$, do you have a name for "the $(k,\epsilon)$ such that there exists a shell $\sigma$ around $\nu$ where $a = a_\epsilon^{(k)}$ in $\sigma$"? (this may not be unique in general, but I'm generally making assumptions which implies that it is unique if it exists).

Amar Hadzihasanovic (Jul 12 2022 at 16:42):

Oh, I got the indexing wrong in my reply, in fact I would use the same as you and have the $a_\alpha^{(k)}$ be $(k+1)$-dimensional.

Amar Hadzihasanovic (Jul 12 2022 at 16:45):

But no, I don't have any name for the individual parts of the shell, as far as I'm concerned you're free to pursue the “quantum number” story ;)

Amar Hadzihasanovic (Jul 12 2022 at 16:46):

(And yes, I do allow for the case that there are already some $n$-dimensional cells around)