Aaron Kriegman /
Correspondence
$\gdef\coker{\operatorname{coker}} \gdef\im{\operatorname{im}} \gdef\coim{\operatorname{coim}}$
Functions are the most fundamental atom in the semantics of mathematics, the tool we use to express all other ideas. At least, this is how we are all taught to view mathematics. But I think things don’t have to, and probably shouldn’t, be this way. Functions allow us to write down well defined expressions, but there are many situations where functions are innelegant and we have to go to great lengths to get around their deficiencies. I think a more general notion of correspondence is needed. In this essay I will show some of the problems with functions, and then start to develop the ideas of an alternative theory. Note that I will assume an undergraduate level of familiarity with math in this essay.
A function can be thought of as a relation between two sets, called the domain and the range, where each thing in the domain is related to exactly one thing in the range. This allows us to write down unambiguous single valued expressions, but as we will see this condition is often unnatural. The most familiar examples of single valued functions being a burden are probably the square root (particularly the complex case) and the complex logarithm. Often when people talk about these functions they will start by specifying exactly which branch they are talking about, when really what they’re saying applies equally well to all branches and this pedantry adds nothing to the conversation. Then in order to make sense of multivalued functions they will do grotesque things like define a sheaf.
My favorite example of the deficiency of functions is angles. We are taught that angles are real numbers which can naturally be added and subtracted, subject to an equivalence modulo some period. There are also hyperbolic angles which are less familiar, but are also conventionally thought of as real numbers that can be added and subtracted. However, if you dive deep enough into hyperbolic geometry, you will find that the space of hyperbolic angles naturally has a second sheet, that it should really be thought of as two copies of $\mathbb R$. That is, it’s naturally shaped like a hyperbola. And the space of elliptic angles (ie angles) is naturally shaped like a circle because of that periodicity.
For these reasons I think that the spaces of elliptic and hyperbolic angles should instead be defined to be the groups $SO(2)$ and $SO(1,1)$ respectively. But then how do we conceptualize those real numbers that we usually use for angles? One could say that the word “angle” means those numbers, although for our purposes I’ll take “angles” to be these underlying groups. The tool we are given to describe this type of correspondence is a homomorphism, which in this case will be between $\mathbb R$ and $SO( * )$ where $ * =2$ or $1,1$.
But which way does the homomorphism go? First let’s try $\mathbb R \to SO( * )$. This is surjective but not injective in the elliptic case, and injective but not surjective in the hyperbolic case. The problem is this fails to describe the second leaf of the hyperbolic angles. If we go the other way $SO( * ) \to \mathbb R$, then we’re surjective not injective in the hyperbolic case, and it just doesn’t work in the elliptic case.
So what should we do? Do we say that elliptic angles have a homomorphism $\mathbb R \to SO(2)$ and hyperbolic angles go the other way $SO(1,1) \to \mathbb R$? I think it’s time to introduce a more general notion of homomorphism, built on top of a more general notion of function.
Correspondences
We’ll call them correspondences. A correspondence is a relation such that if $a\sim x$, $b \sim x$, and $b \sim y$, then $a \sim y$. In other words, in the diagram
flowchart LR
a & b --- x & y
if three of the edges hold, then the fourth holds as well. Here these edges represent two elements being related.
The domain and range will be indicated like $f: A \leftrightarrow B$. Define the image and coimage by $\im f := f(A)$, $\coim f := f^-(B)$.
Note that correspondences are not exactly “multivalued functions”, as the images of two things in the domain cannot overlap, they must be the same or disjoint. Correspondences are geared towards “equivalence-like” relations. A “multivalued function” would just be any relation.
Lemma
A relation $\sim$ is a correspondence between two sets $A$ and $X$ iff it induces an equivalence relation on both the image and coimage, with a natural bijection between the equivalence classes.
Proof
lol no
We will use the familiar notation from functions, but the meaning will now be somewhat different. If $f$ is our correspondence, then $f(a)$ can be thought of as the set of things corresponding to $a$, called the image of $a$. Or sometimes, a statement involving $f(a)$ may implicitly mean “for any $x \in f(a)$.” I’m still working out the details for this part.
Now we can introduce our generalization of homomorphisms. A homospondence is a correspondence $f$ between groups satisfying $f(ab) = f(a)f(b)$ and $1 \in f(1)$. To be less ambiguous, if two of the statements $a \sim x$, $b \sim y$, and $ab \sim xy$ hold, then the third does as well, and $1 \sim 1$. Or, if we interpret $f(a)$ to mean the image of $a$, then $f(a)f(b)$ can be interpreted as a multiplication of sets, $AB := \{ab : a \in A, b \in B\}$.
As an aside, I hear people say thing like “The cute way to show $aNa^- = N$ iff $aN = Na$ is to multiply on the right by $a^-$, but you can’t actually do that, and the actual proof is more rigorous.” But you actually can! Subset multiplication is associative, and if you treat elements as singletons then there’s only one operation being used in expressions like $abAB$.
Now we can generalize the theory of homomorphisms. Define $\ker f := f^-(1)$, $\coker f := f(1)$, or equivalently $\ker f$ is all the things equivalent to $1$ in the induced equivalence relation. Note that a homospondence is a homomorphism iff it’s a function iff the cokernel and coimage are trivial.
Lemma
A homospondence is always $m$ to $n$, $\exists m, n$. In other words, $\#f(a) = n \ \forall a \in \coim f$, $\#f^-(x) = m \ \forall x \in \im f$, where $m, n$ depend only on $f$ and may be infinite.
What we really should be saying actually is that every preimage is a coset of the kernel and every image is a coset of the cokernel. This is a more powerful statement and is how we would prove the lemma.
Lemma
$\ker f$ is a normal subgroup of $\coim f$
Note that $\ker f$ is not generally a normal subgroup of $G$. Take for example the homospondence $f: S_3 \leftrightarrow \{1\}$ where $f^-(1) = S_2$. Also note that as an immediate corollary $\coker f \lhd \im f$.
Proof
Take $a \in \coim f$, so $f(a) = x$. $a(\ker f)a^- = f^-(x)f^-(1)f^-(x^-) = f^-(x1x^-) = f^-(1) = \ker f$.
Theorem (First Isospondence Theorem)
$$\frac{\coim f}{\ker f} = \frac{\im f}{\coker f}$$
We get the first isomorphism theorem as a special case when the coimage is the whole group and $\coker f = 1$, ie when $f$ is a homomorphism.
This theorem can be thought of as giving a sort of greatest common divisor of groups. That is, given a homospondence $f:A \leftrightarrow B$, there is a group $D$ with homomorphisms from $A$ and $B$ which commute with $f$, and such that whenever we have homomorphisms from $A$ and $B$ to some third group they factor through $D$.
I conjecture that there is also a sort of lowest common multiple of groups. That is, a group $M$ with homomorphisms to $A$ and $B$ which commute with $f$ and such that whenever a third group has homomorphisms to $A$ and $B$ they factor through $M$.
Exercise
Apply these ideas to the example of angles. The correspondence should be $\infty$ to $1$ in the elliptic case and $1$ to $2$ in the hyperbolic case, or vice versa. In each case either $f$ or $f^-$ is a function, but now we can have the correspondences pointing the same way.