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If I’m not mistaken, the nLab doesn’t have a good notation for the 2-category of locally small categories. We have Cat for the 2-category of small (or maybe essentially small) categories, and we have CAT for the very large category of large categories (not necessarily small hom-sets). The thing I want is intermediate. I suppose one might use $Set$-$CAT$ (like $V$-$Cat$ for $V = Set$, but allowing large classes of objects), but I find that notation unpleasant. Any suggestions?
You could use UCat where U denotes either a universe or the forgetful functor from 2-categories to 1-categories. The french notation is also alright, where the 2-category of categories is written as Cat with an underline.
How about Cat? I admit it’s a pain to type and it doesn’t work in iTeX math mode…
Edit: I tried Unicode entities for the small caps, and it looks fine outside of math mode (Cᴀᴛ) but inside of math mode the entities render differently somehow: $\mathrm{C}ᴀᴛ$.
Thanks, Harry. Possibly I’ll use one of the two suggestions (although I hope to hear from others as well), starting with a creation of a page to introduce the notation. Would you be able to point me to a reference where either of those is used specifically for the 2-category of locally small 1-categories?
“Others” including Mike, who responded while I was typing. (Thanks for the suggestion, Mike – still weighing options. To my aging eyes, it looks very similar to $CAT$ which is notation used for something else.)
Without being familiar with all the words that French universe-people use, to me, writing “UCat” when U is a universe would make me guess that it means the (2-)category of U-small categories, not U-locally-small ones. And one problem with underlines is that we can’t make an nLab page named $\underline{Cat}$.
The only other thing I can think of is something like $CAT_{ls}$, which maybe isn’t much better than Set-CAT. If I were writing a paper in which I needed the 2-category of locally small categories, but not the 2-category of all large categories, then I would be inclined to denote the former simply by CAT; but I guess we need a more globally consistent notation on the nLab. Would it work to say at the beginning of a particular page “On this page we write CAT to refer to the 2-category of locally small categories” and add a remark to CAT saying that sometimes it is used in that way?
What about $LS Cat$?
Thanks, Toby – I think so far $LSCat$ or $LS Cat$ is my favorite. Even though I didn’t say so before, I happen to agree with what Mike said about $UCat$ – it’s apt to suggest the wrong thing.
Don’t some people use $cat$ for the (2-)category of small categories, $Cat$ for locally small categories, and $CAT$ for large (i.e., possibly class-sized) categories? Not saying we should follow that here, but rather wondering if my memory here is accurate.
One could of course use $U LS Cat$ when one wishes to specify the universe according to which the categories are locally small. (And if you want the number of objects to be limited by another, presumably larger, universe $V$, then one could do $U LS V Cat$. At this point it’s getting complicated.)
The $cat$/$Cat$/$CAT$ trinity is possible; I might have seen something like that before, but if so, I don’t remember where. I know I’ve seen “$set$” used for the category of small sets, but I don’t remember about $cat$.
LSCat is possibly bad notation, see LS-category
I think Kelly someone used underlined $\underline{C}$ to denote the underlying category of a $V$-category $C$ (or the underlying 1-category of a 2-category, which is just a special case, I suppose).
I’ve not seen Kelly use an underline for the underlying. In the TAC reprint of Basic Concepts, the underlying category is denoted $C_0$. (And I could have sworn that it was actually $C_o$ – that’s an ’o’, not a zero – to denote the ’ordinary’ ($Set$-enriched) category in the original edition from 1980 or so, but my memory could be playing tricks on me again.) But, it’s not very important.
I guess your LS here gives me slight pause. But only slight, especially if we were to use only $LS Cat$ and not LS-category. (This reminds me of an amusing thing that happened at Fadell’s birthday conference at UW-Madison, where I unfortunately didn’t meet Zoran. Jim Stasheff was introducing me to a topology professor, whose name I have forgotten, remarking that I was interested in higher category theory, something like that. The professor’s eyes lit up, and he was all excited to have me talk to his student, who he said was working on the same topic. So we did, but it took a minute or two of mutual incomprehension before it dawned on us that his was a different type of category – the Lusternik-Schnirelmann kind!)
Ah, you’re right. I’ve seen it somewhere, though. But perhaps not common enough to warrant an issue.
What’s wrong with $CAT_{ls}$? Or even $lsCAT$/$ls-CAT$? Without capitalisation, $ls$ would probably not be misconstrued to be ’Lusternik-Schnirelmann’
What’s wrong with $CAT_ls$? Or even $lsCAT$ / $ls−CAT$?
Nothing! :-) Maybe one of those is the best choice after all.
If you really want a hyphen in there (although I don’t like it myself), use ls‑CAT
(in itex) to get ‘$ls‑CAT$’. (In LaTeX, it’s not too hard either, but different.)
And I could have sworn that it was actually $C_o$ – that’s an ’o’, not a zero – to denote the ’ordinary’ ($Set$-enriched) category in the original edition from 1980 or so
My goodness, you’re right! I just went and checked. I never noticed that before. Why do you suppose everyone writes it with a zero nowadays?
My goodness, you’re right! … Why do you suppose everyone writes it with a zero nowadays?
It’s an easy thing to misread I suppose (one expects a zero as a subscript before a lower-case o). It does seem odd that everyone, or at least those who retyped the text, misread or overlooked it, but I guess I believe that before I believe that anyone would intentionally change it (because – as you ask – why a zero? it doesn’t make much sense).
I’ve heard it said that Max Kelly’s eyesight had deteriorated badly in his final years, so probably he wouldn’t have noticed a change in the manuscript. Do you think it’s worth bringing to people’s attention now?
I’ve heard it said that Max Kelly’s eyesight had deteriorated badly in his final years
this is true. See http://permalink.gmane.org/gmane.science.mathematics.categories/3602
because – as you ask – why a zero? it doesn’t make much sense
It’s especially odd because when V=Cat, this functor truncates a 2-category to a 1-category, so one would naively expect a subscript like “1”, not “0”.
I just wasted a bunch of time looking through other papers and here’s what I found:
So it looks like even Kelly was not consistent! I would like to know whether there is some explanation for the varying choices.
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