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Stream: theory: category theory

Topic: diagrammatic vs applicative composition order

Tim Campion (Feb 23 2023 at 13:34):

For composition in an $\omega$-category, I'm currently finding it convenient to write $a \circ_\alpha b$ and $\partial_\alpha a$ for $\alpha \in \mathbb Z$ to be defined as

$\partial_\alpha a = \partial_{|\alpha|}^{\alpha / |\alpha|} a$

$a \circ_\alpha b = \begin{cases} a \circ_{|\alpha|} b & \alpha = |\alpha| \\ b \circ_{|\alpha|} a & \alpha = -|\alpha| \end{cases}$

This has eased a lot of casework in a paper I'm writing, saving me from saying "without loss of generality" or else writing out very similar cases quite a bit. There are two issues though:

1. What should $a \circ_{k} b$ mean when $k \in \mathbb N$? Should it mean composition in diagrammatic order, or in applicative order? My intuition has always been that diagrammatic order is "forward" and applicative order is "backward", so I'm inclined to define $a \circ_k b$ to mean diagrammatic composition, and recover applicative composition as $a \circ_{-k} b$. The counterargument would be that, well, applicative composition is conventionally denoted $a \circ_k b$, so maybe that should be my convention. I'm curious if anybody might have any thoughts on this.

2. In my notation, it's not quite the case that $a \circ_\alpha b$ is defined for $\alpha \in \mathbb Z$, because really I have to consider $0$ and $-0$ as being distinct. This is fine so far as it goes -- it just means that I really have $\alpha \in \{\pm 1\} \times \mathbb N$. But it does raise a question -- maybe I should have $\alpha \in \mathbb Z \setminus \{0\}$ and denote what is usually called $a \circ_k b$ by $a \circ_{k+1} b$? There would be some logic to this -- really globular composition $\circ_k$ has to do with the interaction between $1$ dimension and $k$ other dimensions, so there's some logic to thinking of it as a $k+1$-dimensional phenomenon. But it does fit nicely in the usual convention to say that $\circ_k$ composition has to do with matching up $\partial_k$ and $\partial_{-k}$ of the compositands. Not nearly so nice to have a shift here... Anyway, I'd be curious to hear any thoughts about this too.

Mike Shulman (Feb 23 2023 at 17:47):

I would probably argue that the most ordinary sort of $\circ$ should mean applicative order. It's increasingly common to use a semicolon instead to denote composition in diagrammatic order.

Mike Shulman (Feb 23 2023 at 17:48):

I don't know about the other. I would probably tend to go with $\{\pm 1\} \times \mathbb{N}$, unless you end up doing a bunch of arithmetic on subscripts that comes out looking nicer with $\mathbb{Z}\setminus \{0\}$.

Tim Campion (Feb 23 2023 at 19:15):

Hmm... I guess I see the logic to a semicolon. Although I think it would look pretty ugly to have all over the place, and it's really not great to put a subscript on a semicolon, since it already goes below the bottom of the line... $f ;_k g$
(I suppose what I'm saying is that I really kind of prefer diagrammatic order in my context, so I'd almost rather switch to "usual notation for diagrammatic composition" everywhere rather than use "usual notation with applicative composition" everywhere... and I'm thinking that would not look very good)

Tim Campion (Feb 23 2023 at 19:18):

What about a bullet instead? $f \bullet_\alpha g$ or $f \cdot_\alpha g$

Tim Campion (Feb 23 2023 at 19:19):

for comparison $f \circ_\alpha g$ or $f ;_\alpha g$

Tim Campion (Feb 23 2023 at 19:19):

Actually, I think I like $f \cdot_\alpha g$

Tim Campion (Feb 23 2023 at 19:20):

Maybe even better than $f \circ_\alpha g$

Amar Hadzihasanovic (Feb 23 2023 at 19:50):

What do you think of $f \,\#_k\, g$, which is what Steiner uses?

(I personally prefer the look when you reduce the size of the sharp, so $f \, {\scriptstyle \#}_k \, g$)

Amar Hadzihasanovic (Feb 23 2023 at 19:53):

My second favourite notation is $f\, *_k\, g$ which is what Street used in the oriented simplices paper.

Amar Hadzihasanovic (Feb 23 2023 at 19:56):

The latter is also adopted by most of the French working on $\omega$-categories, e.g. here or here.

Amar Hadzihasanovic (Feb 23 2023 at 19:58):

All of these are for diagrammatic order, by the way.

Amar Hadzihasanovic (Feb 23 2023 at 20:01):

As for your question about $k$ vs $k+1$, I think there is a strong precedent for $k$ and it would be confusing to shift it.

Having both $0$ and $-0$ does not look like a major issue if you want to use your notational shorthand.

Amar Hadzihasanovic (Feb 23 2023 at 20:06):

(For what it's worth, I always prefer to use notation and terminology which has some history, rather than make up my own, even when I don't think it is optimal; to me it feels like a small show of appreciation for work that came before our own.)

Mike Shulman (Feb 23 2023 at 20:34):

If you want to subscript a semicolon, you could use a fatter semicolon: $f \mathbin{⨾_\alpha} g$.

Mike Shulman (Feb 23 2023 at 20:36):

I don't always prefer to use notation and terminology that has some history, but I do agree that in general it's better to stick with what's established, to reduce confusion, unless what's established is extremely bad. My favorite remark along these lines is from Jaap van Oosten: "The only thing worse than bad terminology is continually changing terminology". Cf also https://xkcd.com/927/ -- if everyone invented their own terminology and notation, pretty soon no one would be able to read anything written by anyone else.