The n-Category Café

PS: if you think my previous comment was a distraction, then you’ve proven my point :)

Sorry. I shouldn’t post anything when exhausted and delirious.

I think it is inevitable that things we now consider inevitable will be seen as mere distractions by future generations.

It’s not easy to come up with an example. How about explicit construction of curves (1-dim submanifolds of $\mathbb{R}^2$)? The “Deutsches Museum” in Munich has an impressive collection of mechanical devices that let you draw a plethora of specific curves that I did not even recognize the names of (and I spent a considerable amount of time learning differential geometry).

Can we do better than “Groups are interesting as models of symmetry, manifolds are interesting as models of smooth space, therefore group objects in the category of manifolds will be interesting as models of smoothly varying symmetry?”

Why would you want to do better than that? That sounds to me like asking whether Lie groups could be more inevitable than ordinary groups or manifolds are, which doesn’t seem possible. Unless you extend “Lie group” to refer to arbitrary Dynkin-diagram-like things.

I’m not remotely an expert of Lie history, but wikipedia confirms what I vaguely remembered:

Lie’s idee fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Evariste Galois had done for algebraic equations: namely, to classify them in terms of group theory.

which I hadn’t seen mentioned here as part of the motivation. As such it’s important to note that mathematicians needing to work with concrete differential equations was an important thing in the historical development. This naturally leads one to wonder whether the same impetus for mathematicians developing analytical attacks would have been there had there been some mechanical device capable of numerically solving systems of differential equations at the time to engineers and physicists satisfaction. If not, then they would need to arise from the purely spatial motivation described in David Corfield’s original post. I don’t doubt that this would arise, but I wonder what it would have been.

And there above I quote Connes and Consani talking about

how to recover the field of real numbers from a purely algebraic construction, as the function theory over $Spec(S)$,

where $S$ is the ‘hyperfield of signs’. Natural?

But yes, Dynkin diagrams are everywhere, as Hazewinkel explained.

Not all Lie algebras have Dynkin diagrams, not even all physically relevant ones. Example: Virasoro algebra.

Thanks.

This paper is devoted to multifields. The notion of multifield is an immediate generalization of the notion of field. A multifield is just a field, in which the addition is multivalued. Multifields are very natural and useful algebraic objects. However, to the best of my knowledge, they appeared in literature as late as in 2006 in a paper [6] by Murray Marshall, and yet has to find their way to the mainstream mathematics. Probably, the main obstacle is that a multivalued operation does not fit to the tradition of set-theoretic terminology, which avoids multivalued maps.

I believe the taboo on multivalued maps has no real ground, and eventually will be removed. Multifields, as well as multigroups and multirings, are legitimate algebraic objects related in many ways to the classical core of mathematics. They provide elegant terminological and conceptual opportunities. In this paper I try to present new evidences for this.

‘Natural’, ‘useful’, ‘does not fit to the tradition’, ‘taboo’, ‘legitimate’ – it has all the makings of an interesting case study. Now where to find a grad student to undertake one?

I don’t know what a “group object in Rel” is, because Rel isn’t cartesian monoidal. Is there some categorical way to express whatever version of “inverses” hypergroups have?

I should preface this by saying that I haven’t actually looked at any examples. But anyway…

One observation that might be helpful is that Rel is equivalent to the full subcategory of Sup, the category of suplattices and sup-preserving maps, spanned by those suplattices that are power sets. And the equivalence is even monoidal, so that to give a monoid in Rel is the same as to give a quantale structure on a powerset. Since we seem to be talking a lot about multiplying subsets by each other, and identifying elements with singleton subsets, it seems that the quantale point of view might be helpful.

Note that in some ways, quantales act a lot like rings, with the role of “addition” being played by joins/unions. More generally, suplattices act like “modules,” with the powersets being the “free” ones, so quantale structures on the powerset of $X$ could be thought of as analogous to algebra structures on a vector space with basis $X$, and the powerset-quantale arising from a group (or monoid) as analogous to the usual group algebra. Moreover, the condition on a Rel-monoid that the multiplication is a many-valued-function, rather than merely a relation, translates into saying that if $A$ and $B$ are subsets with $A B = \emptyset$, then either $A=\emptyset$ or $B=\emptyset$ — i.e. the corresponding quantale “has no zero-divisors.”

The condition to be a hypergroup, in quantale language, says that the top element is absorbing. That doesn’t seem to have much analogue in ring theory, where there is no top element since we only have finite sums. It also means these quantales are very different from frame-like quantales, in which the top element is the unit.

And speaking of units, I’m confused. Do you intend a “hypersemigroup” to have a unit? The planetmath page doesn’t mention a unit until it gets down to talking about a “multigroup” – does the hypergroup condition imply some unitality as well as inverses?

Let’s see if Connes and Consani agree with Viro. Viro says

The notion of multigroup appeared in the literature in various contexts, sometimes under other names (such as hypergroup and polygroup).

For him, a set $X$ with a multivalued binary operation $(a, b) \mapsto a \cdot b$ is called a multigroup if

• (1) the operation $(a, b) \mapsto a \cdot b$ is associative;
• (2) $X$ contains an element $1$ such that $1 \cdot a = a = a \cdot 1$ for any $a \in X$;
• (3) for each $a \in X$ there exists a unique $a^{-1} \in X$ such that $1 \in a \cdot a^{-1}$ and $1 \in a^{-1} \cdot a$.
• (4) $c \in a \cdot b$ iff $c^{-1} \in b^{-1} \cdot a^{-1}$ for any $a, b, c \in X$.

Connes and Consani are considering canonical hypergroups. Now we have commutativity, and then expressions of conditions (1), (2) and (3). But now instead of (the commutative version of) (4)

$$x \in y + z iff -x \in - y - z,$$

we have

$$x \in y + z implies z \in x - y.$$

Is this difference something to do with the ‘canonical’ in canonical hypergroup?

A good example to have in mind has underlying set $\{+1, 0, -1 \}$. Then the binary operation is given by the rule of signs

$$1 + 1 = 1; - 1 - 1 = - 1; 1 - 1 = \{+1, 0, -1 \}.$$

Think what you know of the sign of the sum of two integers given their signs. It can be extended to a hyperring and even a hyperfield.