The n-Category Café

I don’t completely understand what you’re saying: for example, your equation involves expressions like $W_f/h$, where I don’t understand what the “$/h$” means.

But this is very interesting. If you explain it, we could perhaps create some topological invariant from a category with weakening structure. Even if not all the quandle axioms hold, there are things you can do with just self-distributivity.

Knotty logic!

Thanks Joachim! For many years I had pinned to my office door a portrait of myself drawn by you when you were very young. I was recognizable by the glasses, beard and blue book in my hand: DeMazure and Gabriel’s Groupes Algebriques. It is still on my bookshelf. Alas, the portrait became too dog-eared to survive the many retrenchments of retirement. Tak igen.

Thanks, Tim! This connection turns out to mean that large cardinal axioms are needed to prove some elementary-sounding results about quandles. Over on G+, David Roberts wrote:

Holy cow! There’s a theorem about the unboundedness of a reasonable-looking function $\mathbb{N} \to \mathbb{N}$ whose only known proof uses rank-into-rank embeddings. So what? you say. Well, these give the largest large cardinals we know that do not contradict ZFC! (as far as we know).

He cites this comment:

• Adam Goucher, comment on Does an existence of large cardinals have implications in number theory or combinatorics? , MathOverflow.

which says:

There’s an extremely elementary theorem whose only known proof relies on the existence of a rank-into-rank cardinal (basically the strongest large cardinal axiom not known to contradict ZFC).

Let $R_n = \mathbb{Z}/2^n\mathbb{Z}$.The $n$th Laver table is the unique binary operator $\star : R_n \times R_n \rightarrow R_n$ determined by the following conditions:

• $p \star 1 \equiv p + 1$
• $p \star (q \star r) \equiv (p \star q) \star (p \star r)$

Then the function $f_n(q) = 1 \star q$ is obviously periodic with some period $P(n)$ dividing $2^n$.

The existence of a rank-into-rank cardinal implies that $P(n)$ grows without bound.

Reference: Laver, Richard (1995), “On the algebra of elementary embeddings of a rank into itself”

For some mysterious reason I’m just seeing your comment now, Sam. Your comment is fascinating in a couple of ways. First, I’d never seen this derivation of associativity for a self-distributive binary operation with a unit:

$$\begin{array}{ccl} a \cdot (b \cdot c) &=& (a \cdot b) \cdot (a \cdot c) \\ &=& ((a \cdot b) \cdot a) \cdot ((a \cdot b) \cdot c) \\ &=& ((a \cdot b) \cdot (a \cdot 1)) \cdot ((a \cdot b) \cdot c) \\ &=& (a \cdot (b \cdot 1)) \cdot ((a \cdot b) \cdot c) \\ &=& (a \cdot b) \cdot ((a \cdot b) \cdot c) \\ &=& ((a \cdot b) \cdot 1) \cdot ((a \cdot b) \cdot c)\\ &=& (a \cdot b) \cdot (1 \cdot c) \\ &=& (a \cdot b) \cdot c \end{array}$$

I’ll have to stare at this a few days to make sure it’s not a trick. How in the world did you think of this?

Second, while I don’t yet understand the relation between ordered sets and free self-distributive monoids, I can’t help but hope that this is related to what Tim Campion said in this discussion about cardinals and self-distributive binary operations… since cardinals are (certain especially nice) well-ordered sets, albeit typically infinite.

If I may break it down a bit:

$$a \cdot b = a \cdot (1 \cdot b) = (a \cdot 1) \cdot (a \cdot b) = a \cdot (a \cdot b)$$

and similarly

$$a \cdot b = (a \cdot b) \cdot a$$

So now

$$\begin{aligned} a \cdot (b \cdot c) &= (a \cdot b) \cdot (a \cdot c) \\ &= ((a \cdot b) \cdot a) \cdot ((a \cdot b) \cdot c) \\ &= (a \cdot b) \cdot ((a \cdot b) \cdot c) \\ &= (a \cdot b) \cdot c \end{aligned}$$

As for coming up with this …

Andrew wrote:

If I may break it down a bit:

$$a \cdot b = a \cdot (1 \cdot b) = (a \cdot 1) \cdot (a \cdot b) = a \cdot (a \cdot b)$$

and similarly

$$a \cdot b = (a \cdot b) \cdot a$$

Great!

I’m wondering what sort of algebraic structure we really have here. Alissa Crans has called a set with a left-distributive binary operation is a left shelf. We’re starting with a unital left shelf: one that has an element $1$ with $1 \cdot a = a = a \cdot 1$.

But then Sam C showed that such a thing is also a monoid: the binary operation is associative! And you’ve shown that it also obeys

$$a b = a a b$$

(though of course you did this as part of simplifying the proof that the binary operation is associative).

Taking $b = 1$, we see

$$a^2 = a$$

so it’s also a monoid where every element is idempotent.

Now, I’d really be happy if it were a commutative monoid where every element is idempotent, because this would demystify the law

$$a b c = a b a c$$

It would follow simply from

$$a b c = a^2 b c = a b a c$$

In short: any commutative monoid where every element is idempotent is a unital left shelf. But is the converse true?

Can someone deduce the commutative law from the unital left shelf laws? Or can someone find a counterexample: a noncommutative unital left shelf?

To conclude, I still don’t understand what Sam C said about the relation between unital left shelves and ordered sets. (I’ve been asleep for the last 8 hours.)

But a very nice sort of partially ordered set is a inf-semilattice: one where every finite set has a greatest lower bound. Such a thing is secretly the same as a commutative monoid where every element is idempotent… and so, it gives a unital left shelf.

How does this work? Well, suppose we have an inf-semilattice. The greatest lower bound of $a$ and $b$ is an element $a \vee b$. The greatest lower bound of the empty set is a ‘top’ element $\top$, greater than equal to all the rest. And the operation $\vee$ is commutative and associative, with $\top$ as unit. So we get a commutative monoid, and clearly every element is idempotent.

Conversely, given a commutative monoid where every element is idempotent, we can define $a \le b$ to mean $ab = a$, and check that this defines an inf-semilattice.

Apparently there are noncommutative unital left shelves. Last night I emailed James Dolan, and I just read the reply:

JB: Did you know this? Say you have a binary operation that distributes over itself on the left and has an element 1 that’s a left and right unit. Then it’s associative!

JD: not completely sure whether i knew it ….

JB: I don’t know what this means, except of course that having such a unit is a weird thing.

JD: might it be that it means that you’ve got a semilattice, aka acommutative monoid with all elements idempotent? self-distributivity (xy)z=(xz)(yz) follows there from idempotence and middle-four exchange: (xy)z = (xy)(zz)=(xz)(yz).

so can we get back from “self-distributive with 2-sided unit element” to “semilattice”?

hmm, i guess not. z = 1z = (11)z = (1z)(1z) = zz so indeed any element is idempotent, but there seem to be noncommutative examples. so the fact that we don’t seem to be getting just semilattices is making me suspect that i didn’t ever completely work this out before ….

at first i thought that there were examples involving otto’s law xyz=xz, but that seems to be on the wrong track ….

on the other hand, maybe the variety of monoids that you’re getting is precisely lawvere’s “graphic monoids” (satisfying xyx=yx) from that hegelian taco stuff …. which i should probably try to understand sometime, not having understood too much about it back when he talked about it at buffalo …. i vaguely suspect that it’s one of those sorts of things that i didn’t have the prerequisite mind-frame for back then, but might make perfect sense now …. hmm, particularly if there really is some nontrivial relationship to coxeter groups as some of the google links seem to be hinting ….

hmm, i wonder if the murphy monoid is a graphic monoid …..

The Murphy monoid is discussed here:

• James Dolan and Todd Trimble, Buildings and BN-pairs.

and the Hegelian taco is discussed here:

• F. William Lawvere, Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco”, Proceedings of the First International Conference on Algebraic Methodology and Software Technology, University of Iowa, May 22-24 1989, Iowa City pp. 51–74.

Someone should put this latter paper online—I’ve never read it! I think it contains a 2-categorical diagram shaped like a taco; on the $n$Lab it’s summarized thus:

contains the most explicit expression of Hegelian concepts in the context of the presheaf topos on $\Delta_1$ and a footnote on Lawvere’s relation to Hegelian idealism.

Graphic monoids are discussed here:

• Graphic monoid, $n$Lab.

A graphic monoid is one obeying the identity

$$x \cdot y \cdot x = x \cdot y$$

We have

Theorem. A unital left shelf is the same as a graphic monoid.

Proof. On the one hand, Sam C and Andrew Hubery showed above that any unital left shelf obeys the associative law and the graphic identity holds. The graphic identity works like this:

$$x \cdot y = x \cdot (y \cdot 1) = (x \cdot y) \cdot (x \cdot 1) = x \cdot y \cdot x$$

On the other hand, in any graphic monoid, the left self-distributive law holds:

$$x \cdot (y \cdot z) = (x \cdot y) \cdot z = (x \cdot y \cdot x) \cdot z = (x \cdot y) \cdot (x \cdot z) \qquad \qed$$

A commutative graphic monoid is the same as a semilattice. (In a previous comment I was taking this to be an inf-semilattice, but one can simply say ‘semilattice’.)

However, there are also plenty of noncommutative examples. The simplest one has 3 elements, $\{1, \delta_1, \delta_2\}$ with

$$\delta_i \delta_j = \delta_i$$

and the category of presheaves on this monoid is equivalent to the category of reflexive graphs, as mentioned on the nLab.

Let me just say what the 3-element monoid

$$\qquad M =\{1, s, t \}$$

with

$$s t = s s = s$$

$$t s = t t = t$$

has to do with reflexive graphs. My change in notation is supposed to make it more obvious.

Suppose you have a category and you don’t want to talk about its objects, only its morphisms. Then for any morphism $f : x \to y$ we could define its source $s f$ to be the identity morphism $1_x$, instead of the object $x$. Similarly, we could define its target $t f$ to be the morphisms $1_y$.

With these definitions we have two operations that carry morphisms to morphisms, $s$ and $t$, and they obey the rules

$$t s = s s = s$$

$$s t = t t = t$$

You’ll notice the multiplication rules are backwards from what we had above. So, if you think of a monoid as a 1-object category, we’re getting a functor

$$F: M^{op} \to Set$$

sending the one object of $M^{op}$ to the set of morphisms in our category, and $s,t$ to the operations defined above.

This is just a presheaf on $M$.

But in fact none of this requires having a category: it works whenever we have a reflexive graph: that is, a directed multigraph where each node $x$ has a specified edge $1_x : x \to x$$. So, any reflexive graph gives a presheaf on$M$. And conversely, any presheaf on$M$defines a reflexive graph. (We usually think of reflexive graphs as presheaves on a category with two objects,$V$for vertices and$E$for edges. But this category is the Cauchy completion of the monoid$M$, so their presheaf categories are equivalent.)

I think your phrase “exactly intended” is saying what I meant when I said “capture the essence”. Namely, the quandle axioms are just a distillation of the Reidemeister moves.

Richard wrote:

But my point is precisely that the quandle axioms are not a distillation of the Reidemeister moves per se. They are a distillation of how labels of the arcs of a knot diagram must be chosen in order to ensure that the possible labellings of a knot diagram are invariant under the Reidemeister moves.

Okay, that’s right. I wasn’t trying to be very precise; this is more precise.

I don’t know if when you say, “does not seem to have drawn so much attention” you mean so little attention that any example may be of interest, and I have no way of knowing the likelihood of your already knowing about, or being interested in, what I am about to say, but after I read some work of Vaughan Pratt’s pertaining to programming, I poked around looking at some more recent stuff he has done, all beyond my understanding, and I do recall he discusses just such an operation as you are talking about “travel from x to y and keep going the same distance”.

As Gavin hinted, a “symmetric space” is a set equipped with a binary operation that lets you “go from $y$ to $x$ and keep going the same distance”, reaching a point $x \triangleright y$.

I usually call this operation “reflecting $y$ across $x$”. People often call the map

$$y \mapsto x \triangleright y$$

the “inversion symmetry about $x$”.

A good example of a symmetric space is a sphere: there’s an obvious inversion symmetry about any point $x$. It fixes the point $x$, and reflects any other point across $x$.

As Gavin noted, we can take any group and define

$$x \triangleright y = x y^{-1} x$$

If you ponder the formula, you’ll this is a way to reflect $y$ across $x$. Moreover, this operation obeys all the axioms of an involutory quandle:

• left self-distributivity: $x \triangleright (y \triangleright z) = (x \triangleright y)\triangleright (x \triangleright z)$

• idempotence: $x \triangleright x = x$

• involutoriness: $x \triangleright (x \triangleright y) = y$

In my opinion, the concept of involutory quandle is the most general, most beautiful concept of symmetric space! Usually people consider a more restrictive definition where the symmetric space is a manifold. In this definition we start with a Lie group $G$ and a subgroup $H$ that consists of the fixed points of some involution $\sigma : G \to G$: that is, a Lie group homomorphism with $\sigma^2 = 1$. Then $G/H$ becomes an involutory quandle, where we use the involution to define the operation $\triangleright$.

But clearly (to my mind) this should be a theorem about how to construct symmetric spaces, not the definition. And indeed, one can show that an involutory quandle in the category of smooth manifolds arises via this construction… if an extra condition holds: no point sufficiently near $x$ is a fixed point of reflection about $x$.

(For an example of an involutory quandle that doesn’t obey this extra condition, take any manifold and define $x \triangleright y = y$. Then every point is a fixed point of reflection about $x$.)

All this is great if you like differential geometry: symmetric spaces of this sort are wonderful things. But we can easily generalize the usual group-theoretic definition of symmetric space by letting $G$ and $H$ be arbitrary groups and letting $\sigma$ be a group homomorphism with $\sigma^2 = 1$. In this situation $G/H$ always gets the structure of an involutory quandle.

There should be some theorem saying that all involutory quandles obeying some extra conditions arise this way, but I don’t know it.

For details, see:

• Wolgang Bertram, The Geometry of Jordan and Lie Structures, Lecture Notes in Mathematics 1754, Springer, Berlin, 2000.

The relation to quandles is given in Theorem I.4.3.