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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 7th 2011
    • (edited Sep 7th 2011)

    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 SetSet-CATCAT (like VV-CatCat for V=SetV = Set, but allowing large classes of objects), but I find that notation unpleasant. Any suggestions?

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 7th 2011
    • (edited Sep 7th 2011)

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2011
    • (edited Sep 7th 2011)

    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: C\mathrm{C}ᴀᴛ.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 7th 2011

    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?

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 7th 2011

    “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 CATCAT which is notation used for something else.)

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2011

    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 Cat̲\underline{Cat}.

    The only other thing I can think of is something like CAT lsCAT_{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?

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2011

    What about LSCatLS Cat?

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 7th 2011
    • (edited Sep 7th 2011)

    Thanks, Toby – I think so far LSCatLSCat or LSCatLS Cat is my favorite. Even though I didn’t say so before, I happen to agree with what Mike said about UCatUCat – it’s apt to suggest the wrong thing.

    Don’t some people use catcat for the (2-)category of small categories, CatCat for locally small categories, and CATCAT for large (i.e., possibly class-sized) categories? Not saying we should follow that here, but rather wondering if my memory here is accurate.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeSep 7th 2011
    • (edited Sep 7th 2011)

    One could of course use ULSCatU 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 VV, then one could do ULSVCatU LS V Cat. At this point it’s getting complicated.)

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2011

    The catcat/CatCat/CATCAT trinity is possible; I might have seen something like that before, but if so, I don’t remember where. I know I’ve seen “setset” used for the category of small sets, but I don’t remember about catcat.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 8th 2011
    • (edited Sep 8th 2011)

    LSCat is possibly bad notation, see LS-category

    I think Kelly someone used underlined C̲\underline{C} to denote the underlying category of a VV-category CC (or the underlying 1-category of a 2-category, which is just a special case, I suppose).

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 8th 2011
    • (edited Sep 8th 2011)

    I’ve not seen Kelly use an underline for the underlying. In the TAC reprint of Basic Concepts, the underlying category is denoted C 0C_0. (And I could have sworn that it was actually C oC_o – that’s an ’o’, not a zero – to denote the ’ordinary’ (SetSet-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 LSCatLS 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!)

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 8th 2011

    Ah, you’re right. I’ve seen it somewhere, though. But perhaps not common enough to warrant an issue.

    What’s wrong with CAT lsCAT_{ls}? Or even lsCATlsCAT/lsCATls-CAT? Without capitalisation, lsls would probably not be misconstrued to be ’Lusternik-Schnirelmann’

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 8th 2011
    • (edited Sep 8th 2011)

    What’s wrong with CAT lsCAT_ls? Or even lsCATlsCAT / lsUnknown characterUnknown characterUnknown characterCATls−CAT?

    Nothing! :-) Maybe one of those is the best choice after all.

    • CommentRowNumber15.
    • CommentAuthorTobyBartels
    • CommentTimeSep 8th 2011

    If you really want a hyphen in there (although I don’t like it myself), use ls‑CAT (in itex) to get ‘lsCATls‑CAT’. (In LaTeX, it’s not too hard either, but different.)

    • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeSep 8th 2011

    And I could have sworn that it was actually C oC_o – that’s an ’o’, not a zero – to denote the ’ordinary’ (SetSet-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?

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeSep 8th 2011

    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?

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 8th 2011

    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

    • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeSep 8th 2011
    • (edited Sep 8th 2011)

    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:

    • Kelly-Street “Review of the elements of 2-categories” (1974) writes C 0C_0
    • Street “Limits indexed by category-valued 2-functors” (1976) writes |C|{|C|}
    • Kelly “Structures defined by finite limits in the enriched context” (1980) writes C oC_{\mathrm{o}}
    • Kelly “Basic concepts of enriched category theory” (1982) writes C oC_{\mathrm{o}}
    • Blackwell-Kelly-Power “2-dimensional monad theory” (1989) writes C oC_{\mathrm{o}}
    • Kelly “Elementary observations on 2-categorical limits” (1989) writes C 0C_0

    So it looks like even Kelly was not consistent! I would like to know whether there is some explanation for the varying choices.

    • CommentRowNumber20.
    • CommentAuthorDavidRoberts
    • CommentTime7 days ago

    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 ObjObj of objects and a class MorMor of arrows, such that (s,t):MorObj×Obj(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 ObjObj, 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?

    • CommentRowNumber21.
    • CommentAuthorHurkyl
    • CommentTime7 days ago
    • (edited 7 days ago)

    You can replace a category with an isomohpic one whose homsets are disjoint by setting hom˜(x,y)={(x,f,y)fhom(x,y)}\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:xyf : x \to y”, then the proper class of morphisms is given by the predicate “x,y\exists x,y such that f:xyf : x \to y”. Similarly, if you can formulate the predicate “SS is the hom-set of morphisms from xx to yy”, then the predicate f:xyf : x \to y is given by “S:fS\exists S : f \in S and SS is the hom-set of morphisms from xx to yy”.

    • CommentRowNumber22.
    • CommentAuthorMike Shulman
    • CommentTime7 days ago
    • (edited 7 days ago)

    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×ObjSetMor : Obj\times Obj \to Set and we should be able to do the same thing we do with small categories.

    • CommentRowNumber23.
    • CommentAuthorDavidRoberts
    • CommentTime7 days ago

    @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?

  1. 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?

    • CommentRowNumber25.
    • CommentAuthorDavidRoberts
    • CommentTime6 days ago

    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.

    • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTime6 days ago

    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.