On poset-valued sets

On Monday afternoon I had a look at Andrea Schalk and Valeria de Paiva’s paper on Poset-valued sets, which I was reminded of recently by Greg Restall’s survey of relevant and substructural logics.

It gives a recipe for constructing models of linear logic that have specified properties, by starting from a poset that has the required properties and constructing a category based on that. There is something rather mystifying about the construction as it stands, so I wondered whether there’s some way of thinking about it that sheds more light on it. I think there is, and I may as well record it here in case anyone is interested.

Monoidal centres

I was chatting to Craig “Cap. Astro” Pastro yesterday evening over a beer, and he mentioned an interesting question: if you take the monoidal centre of a braided monoidal category, do you get back the thing you started with? We figured out the answer at breakfast this morning, and I’ll explain it in a bit. But first maybe I should explain what the question means.

More on strictification

Another fascinating day! At lunch time Steve Lack told me some interesting things about the strictification operator for bicategories. Apparently section 5.6 of [GPS] is wrong, and Steve reckons Gray is probably not triequivalent to Bicat. (The mistake in 5.6 is in the very first sentence: the trihomomorphism M: B → Gray constructed there is not actually a triequivalence! Although it’s locally a biequivalence by construction, it fails to be triessentially surjective.)

On the other hand, Steve mentioned a sense in which strictification does produce a (Quillen) equivalence: see his paper here. Perhaps this is my cue to finally learn about model categories…

Calgary 2

I’m in Calgary now, at the CMS meeting. This morning’s session was very interesting, but so far I’ve learnt more from chatting with the other participants. The Sydneyites tell me that Ross Street thinks he has found a good way of thinking about my compact closed biproducts theorem. Unfortunately none of them can remember the details! I’m surprised at how many (very clever) people claim to have had trouble understanding that paper: I thought I’d made it really, really simple, but maybe I’ve confused people by putting in too much detail!

I’m drinking A&W root beer, which I am assured is a famous Canadian delicacy and tastes of sugary toothpaste. Earlier I was stopped by two burly policemen on mountain bikes, who asked me very sternly why I had crossed the (untrafficked) street. I resisted the temptation to give the obvious answer, and engaged the hapless foreigner defence. It seemed to have the desired effect, though the policeman made it clear that only thanks to his boundless magnanimity was I to be spared the $172 fine. North America is a strange place, where pedestrians are forbidden to impede the hypothetical progress of imaginary cars.

I’m speaking at 8:30 tomorrow morning, so I’d better get back to figuring out what to say. (The fear of the unknown has evaporated now I’m here, so I only have to worry about making it comprehensible and engaging…)

On semi-strictness

Update: One correction and one addition; changes in red.
Second update: another correction.

I had a couple of very interesting responses to my speculation about n-categories. As well as John Baez’s comment, I had an email from Tom Leinster pointing out his Structures in higher-dimensional category theory, whose section III.7 contains a similar conjecture.

Thinking about what John wrote, I realise that I made a mistake. I was wrong to claim that the full sub-tricategory of Bicat determined by the 2-categories is a Gray-category: the problem is that post-whiskering with a pseudo-functor is not 2-functorial, even when the objects are 2-categories. (It looks as though Tom’s section III.3 contains the same error? Note that the 1-cells of his 2-Cat are pseudo-functors rather than 2-functors.)

Calgary

Next week I’m off to Calgary, where I’m giving a talk at the category theory session of the Canadian Mathematical Society’s summer meeting, and then a (completely different) talk at FMCS. I’m not too worried about the FMCS talk, because I’ve been to a few theoretical computer science meetings before so I reckon I know roughly what to expect. My topic is reassuring too: it’s simple enough that I’ll have time to explain it pretty carefully in half an hour, yet unexpected enough to be interesting, and reasonably relevant to at least one area of computer science.

The CMS meeting is a different matter. I’ve never even attended a purely mathematical conference, and I confess I’m feeling rather daunted at the prospect of talking at a meeting that includes so many eminent category theorists who I’ve never met. Also my topic is a bit more off-the-wall, though at least there’s a bit more substance to it now than when I wrote the abstract: I’ve worked out a two-dimensional version, which in the special cases where it applies makes it possible e.g. to describe a symmetric monoidal bicategory both rigorously and concisely. I’m working on the slides for the talk right now, but sadly I won’t have time to say much about the bicategory version. (I expect I’ll write a paper about it at some point, where I’ll be able to give all the details.)

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 [B^op, 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.]

Unbiased bicategories and canonical choices

One of my favourite recent books is Tom Leinster’s Higher Operads, Higher Categories. Among the things I like about it is that he’s not afraid to dwell on issues that are conventionally dismissed as trivial, such as the distinction between classical and unbiased monoidal categories and bicategories. The more I think about it, the more I think that the distinction deserves attention: perhaps I’ll return to that in a future post.

There’s a brief remark on p. 123, where he mentions one of the benefits of unbiased bicategories as compared to classical ones. Any bicategory B has a hom pseudo-functor B^op×B→Cat; but defining its action on 1-cells requires a non-canonical choice, when working with classical bicategories. The pseudofunctor should take a pair (f,g) of arrows to the function p ↦ gpf, but classically there is no ternary composition, so we are forced to make an arbitrary choice to bracket this as either g(pf) or (gp)f.

Recently I needed to define the curried version of this pseudo-functor, which goes B^op→[B,Cat]. I’m working with classical bicategories, but I didn’t have to make any arbitrary choices in the definition. (Try it!) That didn’t surprise me at the time, but in the context of Leinster’s remark it seems distinctly strange. It seems that the mere act of uncurrying has introduced some arbitrariness! And sure enough, if you try to define a biequivalence

Bicat(A×B,C) ≈ Bicat(A, [B,C]),

some arbitrary choice does seem to be needed to go from right to left. Suppose we have a pseudofunctor F: A→[B,C], and denote its uncurried version F’: A×B→C. How should we define F’(f,g), for arrows f: W→X and g: Y→Z? Clearly it must be either F(X)(g)⋅F(f)_Y or F(f)_Z⋅F(W)(g), but neither choice is canonical. There is an invertible 2-cell F(f)_g between them, so in a sense it doesn’t matter which we choose, but still one of them must be chosen.

And it matters not a jot whether the bicategories are classical or unbiased; the problem stems from the way pseudo-natural transformations are defined. In the special case of the hom pseudofunctor, ternary composition provides a way out, but in general the difficulty remains.

Hello, world!

I thought it would be fun to start a “work” blog, where I can talk about things that I’m working on or would like to work on, or just think are cool and interesting. It remains to be seen whether I’ll keep it up, so I’m starting out here on wordpress.com. I’ll move to a proper host later on, if that still seems like a good
idea.

My real reason for doing this is to have a forum for speculation. Mathematicians in general are reluctant to speculate, and wary of looking stupid. I don’t want to look stupid either, of course, but refusing to talk about things I don’t understand strikes me as too high a price to pay. (Also I seem to end up making a fool of myself in any case, so I may as well make a virtue of it. :-))

So this is intended to be a sort of public notebook, a repository for ideas that are too trivial or speculative to publish, and which I would otherwise just write on scraps of paper or forget about altogether. It will be interesting to see whether anyone reads it. I would love to read something of the sort written by someone else: am I the only one?

So welcome to bosker blog! Feel free to drop me a comment if you’ve stumbled across it somehow and you want to say hello.