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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.
I have a technical question about the definition of large but locally small categories when working over ZF(C). One could define such a thing as a class $Obj$ of objects and a class $Mor$ of arrows, such that $(s,t)\colon Mor \to Obj \times Obj$ has sets for fibres. Or one could define it as set-enriched category with a proper class $Obj$, where there is given a hom-set for each ordered pair of objects. I assume we can take these hom-sets disjoint (there was a discussion about this at some point, I can’t recall where), but the bigger problem is going from the second description to the first. Only if the hom-sets are defined in a uniform way do we get a proper class containing all the morphisms. Is there any risk, if you like, of a large category of interest falling through the gaps? That is, we have a large category such that we cannot gather all its morphisms into a single class?
You can replace a category with an isomohpic one whose homsets are disjoint by setting $\widetilde{hom}(x, y) = \{ (x, f, y) \mid f \in \hom(x, y) \}$.
As for your question, I presume the devil is in the details of being “given a hom-set”. But if you can formulate the predicate “$f : x \to y$”, then the proper class of morphisms is given by the predicate “$\exists x,y$ such that $f : x \to y$”. Similarly, if you can formulate the predicate “$S$ is the hom-set of morphisms from $x$ to $y$”, then the predicate $f : x \to y$ is given by “$\exists S : f \in S$ and $S$ is the hom-set of morphisms from $x$ to $y$”.
If we work in NBG, which is basically the same as working in ZF(C) but with more reification, then we would have an actual proper-class-function $Mor : Obj\times Obj \to Set$ and we should be able to do the same thing we do with small categories.
@Mike, sure, but I was checking ZF(C). I’m making sure my approach works in multiple systems.
@Hurkyl, thanks, I get the general idea. I was wondering about what one might call the ’usual’ large categories that arise in practice: algebras for a(n essentially) algebraic theory or a monad, or an operad, or sheaves or …. I don’t think there’s any risk, but wanted to check. Presumably if one has it for Set then the rest follow?
I always assumed that when people talked about classes in ZFC they were implicitly using NBG. There’s no such thing as a class within ZFC! Am I missing something?
You can treat a class as a syntactic object, but considered up to extension, I guess. A formula is not an object of ZFC, but they can be manipulated with similar operations as for sets: Cartesian products, disjoint unions (both finite), functions, quotients by equivalence relations (using Scott’s trick if needed). Maybe it’s just people working with the model of NBG that arises from the sets and classes of ZFC, so only classes defined by formulas, not “arbitrary” classes.
Maybe it’s just people working with the model of NBG that arises from the sets and classes of ZFC
Yeah, my understanding is that those are basically equivalent. So if NBG “works”, then so does ZFC by using this model.
An object $A$ in a category $C$ is called a locally small object if the slice category $C/A$ is essentially small (has a small skeleton). In the setup of module categories and abelian categories a stronger notion is (widely) called a locally small category, namely a category such that all its objects are locally small. That is, subobjects of any object $A$ (isomorphism classes of monomorphisms with target $A$) form a set. E.g. 5.23.2 in
and 1.2 in
I do not know how to nlabify this. It is important as it is quite standard in that subject.
P.S. I understand that locally small categories are also called well-powered in nonabelian contexts.
Jeremy Rickard wrote at stackexchange/abelian-categories-with-generator-objects-are-locally-small
Using “locally small” for what is now usually called “well-powered” wasn’t particularly idiosyncratic for the time. It was once quite common terminology and used in some well-known early books such as Mitchell’s “Theory of Categories”. In Mac Lane’s “Categories for the Working Mathematician” he explicitly states that he avoids using the term “locally small” at all because of the two conflicting meanings.
Working mathematicians in module categories and abelian categories are still often using this terminology, especially when saying locally small abelian category.
“Well-powered” means that the posets of subobjects are small, which is a much weaker statement than that the whole slice category $C/A$ is essentially small. Did you really mean the latter?
That’s the same mistake that yesterday I had fixed in another entry: here. (In the reference which Zoran cited there it is stated correctly.)
Oh, that is my mistake, of course I meant monomorphisms (I was lost once in formalizing that the simplest way). Though I dislike calling it poset if it is a class to start with and specially if one does not take the equivalence classes but the category where there are possibly equivalent ones (hence a preorder rather than a partial order),
There is an entire circle of entries related to localization in Abelian categories which I am trying to improve these days, and for next few days please consider them as work in progress till certain batch of work is finished. The main problem as usual is that this is an area where there is plethora of (equivalent or equivalent under some additional conditions) formalisms and competing or contradictory terminologies in the literature. One of the reasons people often rediscover known things. I am reviewing one paper in Mathematical Reviews on the same topic and also revising a preprint (a joint work) with slight variant on the topic. Not only that the entire paper under MR (published in Comm. Alg.) consists of results known for 50 years but it is not a single excursion of the authors, they have an earlier work in 2014 or so where they have another half of the work, so after 8 years nobody told them that it is known, and Zbl review does not note either (I wrote to the Zbl author, no response, people just write surface level reviews and do not pay attention to the published literature).
I would say “the poset of subobjects is small” or “the preorder of monomorphisms is essentially small”. That is, “a subobject” is an isomorphism class of monos with fixed codomain.
Yes, I tend to think of subobject as an isomorphism class but it is hard, in my experience, to impose that discipline on the community.
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