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Stream: practice: terminology & notation

Topic: Representing object for forgetful functor

Ralph Sarkis (Nov 29 2023 at 16:36):

What is your preferred notation for the representing object of a forgetful functor (in particular to $\mathbf{Set}$) ?

I think $\mathbf{1}$ would be great because morphisms $\mathbf{1} \to X$ would be the same thing as elements of the underlying set of $X$, but $\mathbf{1}$ is already the terminal object which does not necessarily represent the forgetful functor (e.g. in $\mathbf{Grp}$ where $\mathbf{1}$ is also initial).

Chris Grossack (they/them) (Nov 29 2023 at 16:48):

I use $F1$ a lot, even if a free functor doesn't actually exist. Also I've seen notations like $\mathcal{C}[X]$ based on the analogy that $R[X]$ represents the forgetful functor $R\text{-alg} \to \mathsf{Set}$.

Todd Trimble (Nov 29 2023 at 17:09):

FWIW, I typically just use "$U$" for "underlying" (functor). Is that unimaginative? Yes. So then, maybe it's $U$ for unimaginative.

I can't decide if it's a nitpick or not, but there is a point to considering any functor $C \to D$ as a "forgetful" functor, i.e., as forgetting something, even if that something is nothing (equivalences don't forget anything definable in the language of category theory). Analysis of this point (what type of thing is being forgotten?) is what led to the theory of stuff, structure, property.

Nathanael Arkor (Nov 29 2023 at 17:09):

A presheaf $U \colon \mathcal C^\circ \to \mathbf{Set}$ is representable iff it admits a left adjoint relative to the functor $\mathbf 1 \to \mathbf{Set}$ picking out the terminal object. It seems reasonable then to call this relative left adjoint $F \colon \mathbf 1 \to \mathcal C^\circ$, and hence to call the representing object $F1$.

Though I think the question is similar to "What name should I give to a left adjoint?". Often you have an explicit description of the left adjoint/representing object, and this suggests an informative name, rather than using the same symbol for every left adjoint/representing object.

Nathanael Arkor (Nov 29 2023 at 17:11):

Todd Trimble said:

FWIW, I typically just use "$U$" for "underlying" (functor). Is that unimaginative? Yes.

I think "mnemonic" is a more suitable adjective than "unimaginative" :)

Ralph Sarkis (Nov 30 2023 at 07:14):

Thank you all, I do like $F1$, but I tried merging them together and we'll see how I feel about it (ignore the colors):
image.png

Nathaniel Virgo (Nov 30 2023 at 07:22):

I think if I saw that I would assume it was a weird H or A

Ralph Sarkis (Nov 30 2023 at 07:28):

Yeah same (I don't think that is a problem).

John Baez (Nov 30 2023 at 08:32):

I use $F$ for "free" and $U$ for "underlying" (aka "forgetful") for left and right adjoints when the free-forgetful viewpoint is helpful, or $L$ and $R$ for left and right adjoints when I'm trying to take a less evocative, more abstract approach. I think these are pretty common, so easily understood.

John Baez (Nov 30 2023 at 08:33):

It's unfortunate that "free" and "forgetful" both begin with $F$ in English... but anyway, I never use $F$ for "forgetful".

Morgan Rogers (he/him) (Nov 30 2023 at 10:10):

Ralph Sarkis said:

Thank you all, I do like $F1$, but I tried merging them together and we'll see how I feel about it (ignore the colors):
image.png

What's wrong with just using "F1", as long as you explain the context for that choice of notation?

Ralph Sarkis (Nov 30 2023 at 11:08):

Nothing wrong, I am trying out the merged symbol in case I prefer it.

Mike Shulman (Nov 30 2023 at 17:04):

It's a noble experiment, but personally I think the merged symbol looks pretty ugly.

Nathanael Arkor (Nov 30 2023 at 17:10):

I imagine a "1F" ligature would look much nicer. Arguably the order would be wrong, but one could justify it by asserting that it's in diagrammatic composition order :)

Ralph Sarkis (Nov 30 2023 at 17:13):

image.png

Mike Shulman (Nov 30 2023 at 17:53):

Sorry, I don't like that any better.

Reid Barton (Nov 30 2023 at 19:57):

How do you pronounce this?

Ralph Sarkis (Nov 30 2023 at 20:08):

"fwan" maybe (like swan but with an f)

Nathaniel Virgo (Dec 01 2023 at 02:05):

Part of the problem is that the F is italic while the 1 is upright. You can fix that with textit, e.g. F\hspace{-3pt}\textit{1}
image.png

or \textit{1}\hspace{-2.5pt}F
image.png

I'm unsure if a ligature is the best choice of notation but maybe these are more aesthetically pleasing.

Nathanael Arkor (Dec 01 2023 at 08:41):

Nathaniel Virgo said:

or \textit{1}\hspace{-2.5pt}F
image.png

This is much closer to what I had in mind (though the question still remains about whether it is preferable to use such a symbol) :+1:

Matteo Capucci (he/him) (Dec 01 2023 at 11:35):

Call it a 'fun' object

Damiano Mazza (Dec 01 2023 at 11:59):

That joke might already be copyrighted by French-speaking people working on the "field with one element" $\mathbb F_{\mathrm{un}}$ :wink: