Speculating about n-categories

According to the stats I’ve already had one reader, so perhaps it’s time for me to stick my neck out: this speculation tag is intended for things that might be utter nonsense, a fair proportion of which I shall no doubt be embarrassed to have written once I’ve realised what rubbish they are. The hope, of course, is that one or two might turn out to contain a useful idea. We shall see.

The theory of weak n-categories is very much in its infancy. (This recent-ish project proposal paints a helpful picture of what has been, and what remains to be done.) A mature theory of n-categories ought at least to have definitions of ‘n-functor’ and the n different kinds of ‘transformation’ needed to define an (n+1)-category of n-categories. I’m not aware that any of the proposed definitions has such a thing, though I must confess that I have read only a small fraction of the relevant literature, so I could be mistaken in that. Once that is done, there is an obvious desideratum that I haven’t seen mentioned: a Yoneda theorem, which at the very least should show how, for any (n+1)-category C and (n+1)-functor F: C → n-Cat, to exhibit an appropriate equivalence

FA ≈ (n+1)-Cat(C, n-Cat)(C(A, -), F).

I would imagine that such a theorem, together with a few supporting lemmas, would fairly readily yield a coherence theorem of the form “every weak n-category is equivalent to a semi-strict one”, along with a specification of what it means for an n-category to be semi-strict. It certainly works in dimensions two and three, as follows: the bicategorical Yoneda lemma shows that every bicategory B is biequivalent to its image in [Bop, Cat], which is a 2-category (since Cat is). In dimension three, the tricategorical Yoneda lemma (I should really say conjecture, since I don’t think there is a full proof yet) would show that every tricategory is triequivalent to one that is as strict as the full sub-tricategory of Bicat determined by the 2-categories. How strict is that? Well, I can easily convince myself that it’s a Gray-category. It is surely not much of a stretch to conjecture that every weak 4-category is equivalent to one that’s as strict as the full sub-4-category of Tricat determined by the Gray-categories. It can’t be hard to figure out just how strict that is, can it? I think it’s locally a Gray-category, and horizontal composition is strictly associative with a strict unit, and is also strict in each argument separately.

[See here for an  update.]

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5 Responses to Speculating about n-categories

  1. John Baez says:

    Hi –

    Glad to see you’re blogging!

    You’ve got my curiosity up, but there’s no link here:

    The theory of weak n-categories is very much in its infancy. (This recent-ish project proposal paints a helpful picture of what has been, and what remains to be done.)

    Speculation is a great thing when done responsibly… i.e., when
    one does ones damnedest to take into account all the facts and patterns.

    Both James Dolan and I, and Ross Street and Brian Day, have broken our lances trying to create a general theory of “semistrict n-categories” based on a recursively defined notion of “Gray tensor product” for semistrict n-categories. Our approaches were very similar; Day and Street called their gadgets “files”. Both our approaches break down right at tetracategories, and for the same reason – which I forget. It was a long time ago.

    So, in fact it’s much better to do as you suggest: look and see how the “nice” tricategories – namely Gray-categories – organize themselves into a “nice” tetracategory, and then make this into a definition of “semistrict tetracategory”.

    However, I don’t think you should look at the full sub-4-category of Tricat determined by the Gray-categories. After all, that’s not how it works one step down. One looks, not at the full sub-3-category of Bicat determined by the (strict) 2-categories, but at:

    [2-categories, strict functors, pseudonatural transformations, modifications]

    This is what gives a Gray-category – the one called Gray!

    (I keep asking if anyone has proved that Gray is triequivalent to Bicat – surely this must be the hope – but I must be asking the wrong people, because nobody has confessed to knowing the answer.)

    I don’t know if someone has found the “nice” tetracategory whose objects are Gray-categories.

    Btw, you really should read Nick Gurski’s thesis Algebraic Tricategories when it comes out – he said he’d put it on the arXiv.

    Best,
    jb

  2. bosker says:

    Wow! It’s scary how quickly you found my blog. Thanks for pointing out the missing link. I’ve fixed it, though I doubt you’ll learn much from it, since I assume you had a hand in writing it.

    On speculation, it sounds as though you’re talking about the respectable kind of speculation, the sort that is written by bona fide experts and sometimes published in proper journals. (Your stabilization hypothesis is a fine example.) Of course I agree that this sort of speculation is wonderful, and I wish there were more of it. But I was also talking about the less respectable kind, that may not be sufficiently well-informed and which may be therefore be obviously wrong (or alternatively obviously trivial) to someone better-informed. This sort is dangerous, both because it can make the author look stupid and because it can be an excuse for intellectual laziness. Perhaps it’s simply a bad idea; I’m toying with the notion that maybe it isn’t so bad after all, provided it doesn’t pretend to be other than it is.

    Now I’ve written all this waffle, and I haven’t addressed the serious part! It’s very interesting to learn that generalising Gray-categories doesn’t work in dimension four. (If you ever remember why not, I’d love to know.) In the morning, before I head to Manchester, I’ll write something about the real substance of your comment, maybe as a separate entry.

    Thanks for letting me know about Nick Gurski’s thesis. I look forward to reading it!

  3. Welcome to the blogosphere. Published mathematical speculation, as you suggest in another post, is far too rare. I believe that this is a part of an unhealthy unwillingness to reveal one’s deeper thinking.

    I doubt John’s intending anything as restrictive as you suggest concerning speculation. After all he’s participating in this piece of speculation with a miserable philosopher. I like his choice of the term ‘responsibly’. It chimes with the philosophy of a rather neglected thinker called Michael Polanyi. The notion of the responsibility of a scientist allows him to isolate something universal in the thinking of a particular scientist, despite the specificity of their location in time, training, etc.: “Anyone in my situation should act like this.”
    Speculation done with an honest appraisal of its likely value in view of what one understands to be one’s level of informedness can’t be harmful.

  4. bosker says:

    Thanks! I’m delighted that you’re here.

    I intended no slight to John. I was just worried that I’m not “[doing] [my] damnedest to take into account all the facts and patterns”, and feeling a little guilty about it!

  5. Pingback: Bosker Blog » Blog Archive » On semi-strictness

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