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Stream: practice: terminology & notation

Topic: how do you denote these things

Matteo Capucci (he/him) (Nov 10 2022 at 12:34):

I like to denote mathematical objects in consistent ways, like any diligent mathematician. For example, all my categories are \mathbf, all my double categories are \mathbb, functors are uppercase Latin whereas morphisms are smallcase Latin, and so on.
Now the trouble starts when you have something that started as an object of some kind, like a double category $\mathbb C$, but then is turned into an object of another kind, like its (single!) category of morphisms $\mathbb C_1$/$\mathbf C_1$.
In your opinion, is it better to preserve the original 'kind' despite the modifier $-_1$ or to switch to the notation of the new kind?

Ralph Sarkis (Nov 10 2022 at 12:49):

I would prefer you don't switch the font.

An alternative that would be as intuitive (for me) is to have $\mathbf{C}_1$ be in the data that defines $\mathbb{C}$.

Dylan Braithwaite (Nov 10 2022 at 13:41):

I agree with the alternative. Something like “Let $\mathbb C = (\mathbf C_1, \mathbf C_0)$ be a double category” seems nicest to me. But I guess this doesn’t work if you want to write something like $\mathbb C_L$ for the bicategory of loose morphisms for example

Matteo Capucci (he/him) (Nov 10 2022 at 13:53):

Ralph Sarkis said:

An alternative that would be as intuitive (for me) is to have $\mathbf{C}_1$ be in the data that defines $\mathbb{C}$.

Yeah that's what I meant. But one might want to write $\mathbb C_1$?

Matteo Capucci (he/him) (Nov 10 2022 at 13:53):

I'm really on the line

Joe Moeller (Nov 10 2022 at 15:39):

My personal opinion is that font is a terrible place to encode critical information. I really think you should just say what you mean. Of course, I've consistently gotten tons of pushback when I bring this up to any coauthor. I use mathsf for any specifically named category-like gizmo: Set, Cat, Prof, Fib, etc. and mathcal for generics like C, D etc. If I'm talking both about the 1-category of categories and the 2-category of categories at the same time, I'd say "Consider the 2-category $\mathsf{Cat}$ of categories, functors, and natural transformations", use Cat_1 for it's truncation, and Cat_h for it's homotopy 1-category.

John Baez (Nov 10 2022 at 16:41):

I like using fonts to encode these distinctions but I think it's always good to also say what you mean in words, repeatedly, not expecting readers to memorize your conventions.

John Baez (Nov 10 2022 at 16:44):

Redundancy is very helpful for clearly conveying information. Some mathematicians ignore this and write as if the only goal was to make it theoretically possible to figure out what they mean, like cracking a code.

John Baez (Nov 10 2022 at 16:46):

I had the problem Matteo describes in my latest paper on double categories.. . and I forget how we resolved it!

John Baez (Nov 10 2022 at 16:47):

It was a real dilemma.

Mike Shulman (Nov 10 2022 at 19:04):

In general I think a good approach is for operations that "change the kind" in this way to be denoted more obviously as functions rather than with a notation like subscripts that tends to disappear visually. For instance, if double categories are in \mathbb and bicategories are in \mathcal, then I would notate the bicategory of loose morphisms of a double category by $\mathcal{L}(\mathbb{C})$. That way it is clear that the object itself is a bicategory, and it is constructed from a double category.

Mike Shulman (Nov 10 2022 at 19:06):

I definitely don't think that the category of morphisms of a double category $\mathbb{C}$ could be denoted by $\mathbf{C}_1$. I would regard that not just as poorly-chosen notation but bordering on incorrect.

John Baez (Nov 10 2022 at 19:30):

Mike will be happy that Christina and Kenny and I wound up agreeing with him:

Conventions

In this paper, we use a sans-serif font like $\mathsf{C}$ for categories, boldface like $\mathbf{B}$ for bicategories or 2-categories, and blackboard bold like $\mathbb{D}$ for double categories. For double categories with names having more than one letter, like $\mathbb{C}\mathbf{sp}(\mathsf{X})$, only the first letter is in blackboard bold. In this paper, 'double category' means 'pseudo double category'. A double category $\mathbb{D}$ has a category of objects and a category of arrows, and we call these $\mathbb{D}_0$ and $\mathbb{D}_1$ despite the fact that they are categories.

Mike Shulman (Nov 10 2022 at 21:29):

To clarify, I'm okay with "let $\mathbb{C} = (\mathbf{C}_0,\mathbf{C}_1)$," just not with changing the font and still thinking of the "C" in $\mathbf{C}_1$ as referring to the same object $\mathbb{C}$.

John Baez (Nov 10 2022 at 22:00):

Hmm, I was not talking about thinking, purely notation. Our dilemma was: if we denote double categories by things like $\mathbb{C}$, how should we denote the two categories they're made of: $\mathsf{C}_0$ and $\mathsf{C}_1$ (since that's our font for categories) or $\mathbb{C}_0$ and $\mathbb{C}_1$ (since they are entities derived from $\mathbb{C}$ via systematic processes).

John Baez (Nov 10 2022 at 22:03):

We wanted a systematic rule so we could say stuff like

Let $\mathbb{D}$ be another double category... then $\mathbb{D}_1$ [or $\mathsf{D}_1$] is cartesian if...

without having to remind the reader, every time, of how we're getting $\mathbb{D}_1$ (or $\mathsf{D}_1$) from the double category $\mathbb{D}$.

Mike Shulman (Nov 10 2022 at 22:24):

Yes, I'm talking about notation too. I think I'm not being very clear. Let me try again.

If I have a function $f:X\to Y$, and I consider the value of that function $f$ at some input $x\in X$, then any notation for that value should include $x$ so that the reader can tell what the function is being applied to. This is obviously the case for the usual function-application notation $f(x)$. It's also the case for other notations such as $x^2$ or $|x|$ or $\sqrt{x}$. It would be wrong to write the value of the function $f$ on the input $x$ with a notation like $\langle y\rangle$, because $y$ is not the object that the function is being applied to; $x$ is.

Similarly, if $f$ is the function that takes a double category to its category of arrows, it would be wrong to write the value of $f$ on an input $\mathbb{C}$ with a notation like $\mathsf{C}_1$, because $\mathsf{C}$ is not the object that the function is being applied to; $\mathbb{C}$ is. In all of my experience of the way that notation is used in mathematics, there is no a priori connection in meaning between variable names that are "the same letter" written in different fonts. The variable $x$ is as different from the variable $X$ as it is from the variable $y$ or $a$; if I want to I can use $x$ to denote a topological space and $X$ to denote a regular cardinal. Similarly, $C$, $\mathbf{C}$, $\mathsf{C}$, and $\mathbb{C}$ are just different variable names, and some object that is denoted by $\mathbb{C}$ cannot suddenly switch to being denoted by $\mathsf{C}$ just because some function is being applied to it whose outputs belong to a type whose elements we conventionally denote with sans-serif letters.

John Baez (Nov 10 2022 at 22:33):

Okay, so we agree: we shouldn't use font change to indicate application of a function or functor to some variable. We decided that principle overrode our desire to use different fonts for different types.

Mike Shulman (Nov 10 2022 at 22:35):

Yes, we agree. I was just clarifying that I didn't object to Dylan's suggestion above.

Matteo Capucci (he/him) (Nov 10 2022 at 22:37):

Mike Shulman said:

In general I think a good approach is for operations that "change the kind" in this way to be denoted more obviously as functions rather than with a notation like subscripts that tends to disappear visually. For instance, if double categories are in \mathbb and bicategories are in \mathcal, then I would notate the bicategory of loose morphisms of a double category by $\mathcal{L}(\mathbb{C})$. That way it is clear that the object itself is a bicategory, and it is constructed from a double category.

Uhm this feels like the most convincing take here... in a sense both options I proposed have flaws: using a subscript or decoration to denote a functor is confusing, and so is using a change in font.

John Baez (Nov 10 2022 at 22:40):

Yeah, that avoid both problems. We just didn't think of that. We might still have decided this notation was too "bulky", since we were talking about things like $\mathbb{C}_0$ and $\mathbb{C}_1$ over and over again, a lot. But luckily our paper is published so we don't have to think about this anymore!

Mike Shulman (Nov 10 2022 at 22:42):

I think denoting a functor by a subscript or decoration can be okay sometimes, just as ordinary functions are sometimes denoted that way. One has to be careful with it, and when you are trying to match fonts to types consistently it does step out of such a system, but it doesn't rub me as wrong the way "denoting a functor with a change of font" does.

Matteo Capucci (he/him) (Nov 11 2022 at 08:38):

John Baez said:

Yeah, that avoid both problems. We just didn't think of that. We might still have decided this notation was too "bulky", since we were talking about things like $\mathbb{C}_0$ and $\mathbb{C}_1$ over and over again, a lot. But luckily our paper is published so we don't have to think about this anymore!

I wouldn't say your choice was bad, actually that's the way I've been denoting them myself all this time. I'm nitpicking to astronomical levels in this thread!

John Baez (Nov 11 2022 at 16:11):

Yes, a stream on a category theory server specially devoted to terminology and notation is a nitpicker's idea of heaven! :innocent: