The n-Category Café

Well, the analogy between universal covering spaces and algebraic (or say separable) closures isn’t perfect. Basically, a separable closure is just a way to package all the finite separable extensions together, so that a closer topological analog would (morally) be a suitable inverse limit over finite covering spaces – and this will be a pretty complicated space, even when you’re starting with the circle.

The passage from the universal cover to this more complicated guy amounts to only remembering the profinite completion of the fundamental group – which is all the algebra can see in general (c.f. etale homotopy theory).

Also worth noting is that, from this perspective, the circle is much closer to a finite field than the rationals: the profinite fundamental group is, in both cases, Z hat.

While I’m making a post, I’ll go off-topic a bit and say that I really appreciate your philosophical contributions to this blog, David, even if I have nothing to say about them.

From Plato to predicativity
“The relationship between mathematics and physical
reality, and that between provability and truth have
been studied from Plato till the present day.”

CAUSALITY: Models, Reasoning, and Inference by Judea Pearl
[A related dialectic?]
“Ten years ago, when I began writing “Probabilistic
Reasoning in Intelligent Systems (1988), I was working
within the empiricist tradition. In this tradition,
probabilistic relationships constitute the foundations
of human knowledge, whereas causality simply provides
useful ways of abbreviating and organizing intricate
patterns of probabilistic relationships. Today my view
is quite different. I now take causal relationships to
be the fundamental building blocks both of physical and
of human understanding of that reality, and I regard
probabilistic relationships as but the surface phenomena
of the causal machinery that underlies and propels our
understanding of the world. Accordingly, I see no greater
impediment to scientific progress than the prevailing
practice of focusing all of our mathematical resources on
probabilistic and statistical inferences while leaving causal
considerations to the mercy of intuition and good judgment.”

GERALD EDELMAN | “The most important thing to understand is
that the brain is “context bound.” It is not a logical system
like a computer that processes only programmed information;
it does not produce preordained outcomes like a clock. Rather
it is a selectional system that, through pattern recognition,
puts things together in always novel ways.”

http://hugo.mercier.googlepages.com/Mercier2005mathematics.pdf
Nonetheless, mathematics relies on cognitive mechanisms that
were empirically acquired during phylogenetic evolution, and
this explains their “unreasonable effectiveness”.

SH: I think that pattern recognition is the evidence our minds
process to establish causality as the underlying shaping metaphor,
of which cultural evolution is a more recent abstract development.
I think the enigma occurs before noticing “unreasonable effectiveness”.

My point of view is just that sometimes you define the base space first and the universal cover second, and other times you do it the other way around. Typically if you need object $A$ to define object $B$, then $B$ is rightfully seen as more complex than $A$.

We need the field $Q$ to define its algebraic closure, but we need the space $R$ to define the circle. In each case, the logically earlier concept seems more simple.

We can come up with an example similar to the Galois one in topology. Let $X$ be two circles glued together at a point. Then I think most people would say that its universal cover is more complicated than $X$ is.

We can do it the other way around, too. It’s possible to start with an easy algebraically closed field $L$ and carve out a subfield $K$ over which $L$ is algebraic. Then the simpler object is $L$ and the more complicated one is $K$

David’s question was “What should one say to that?”

Here is one possible answer: consider the equation x^n + y ^n = 1, with n > 2. If you choose any value of x in the algebraic closure Q-bar of Q, one can obviously find a corresponding y in Q-bar that solves this equation. On the other hand, if we choose x in Q, it is a difficult theorem that we can’t then solve for y in Q (unless we are in a trivial case where x or y equals zero).

This brings out the sense in which an algebraically closed field is simple, while a non-algebraically closed field such as Q is not (despite initial appearances).

This example might seem a little flippant, and I wouldn’t entirely disagree with such a sentiment, but I do think it brings out something important. Algebraic closure is a kind of “simpleness” (in a non-technical sense; I just mean in the everyday English use of the word simple, as in David’s post) for fields, of a very specific nature, just as simple connectedness is a very particular kind of “simpleness” that a topological space can have. But since topology is normally easier to visualize than algebra (especially for non-experts), the “simplicity” of an algebraically closed field such as Q-bar can be hard to appreciate at first (say in comparison to
the evident simplicity of the real line, when contrasted with the more intricate shape of the circle).

Giving the example of a famously difficult Diophantine equation is then a genuinely useful way to illustrate this “simplicity” in the algebraic setting.

This is just a repetition of Matthew’s explanation, but in the context of this cafe, it might be worth spelling out the sense of simplicity involved.

The algebraic closure $\bar{Q}$ is simple in that it is easy to have maps

$Spec(\bar{Q}) \rightarrow Y,$

while maps

$\Spec(Q) \rightarrow Y$

are quite difficult. This is the same for universal covering spaces in topology. In many concrete situations, the awkward notion of a ‘multi-valued map’ from $X$ to $Y$ comes up naturally, which is then clarified by unwinding to a map

$\tilde{X} \rightarrow Y.$

The problem of getting to an actual map

$X \rightarrow Y$

is then one of these problems of descent, in either situation.

Perhaps the kind of complexity that one usually associates with $\bar{Q}$ is actually the price of this simplicity: The automorphism group of $\bar{Q}$ is quite large and complicated. This phenomenon seems to be common in usual geometry as well. At least if one includes the complex structure, then the automorphism group of a compact hyperbolic Riemann surface is finite, while its universal covering space is acted on by $PSL_2(R)$. [I would *not* venture at the moment to argue that this is necessarily more complicated than a finite group, but it is unquestionably larger.]

There is another perspective, similar to that expressed by James, which is a bit hard to make precise: the simplicity of R is deceptive. If we view it in the same ‘bottom-up’ fashion as $\bar{Q}$, it’s horribly complicated.

Thanks everyone for these great comments.

Regarding the ‘imperfections’ of $\mathbb{Q}$, Ittay Weiss was wondering after the talk why it was that polynomial splitting was the only imperfection to be considered. Why not a trigonometric function?

To the example of $sin x = 0$, we can say something. Yves André discusses here the Galois extension brought about by adding $\pi$. Its conjugates could be thought of as all the non-zero rational multiples of $\pi$.

But the list of conceivable imperfections is endless. Why are some so much more interesting than others?

As Minhyong wrote, studying algebraic extensions of Q is analogous to (in topology) studying covering spaces of some
base space X, which are the same as fibre bundles with discrete fibres. As he goes on to remark, lots of fibre bundles with non-discrete fibres are also of much interest.

Thus, one would expect there to be an interesting analogy back on the Q side of things, and indeed there is: if we consider a curve (e.g. y^2 = x^3 - x) with coefficients in Q, we can think of this as a fibre bundle of (in this case, genus 1) curves over Q.

(The analogy on the topological side would be to replace Q by some space, e.g. the line, with coordinate t, say, and then to consider a family of curves over the line, e.g. y^2 = x^3 - t x.)

If one focuses attention on a generic point of y^2 = x^3 - x, one rapidly discovers a field of transcendence degree 1 over Q (in this case, the field Q(x)[y]/(y^2 - x^3 + x) ), and indeed, studying such field extensions of Q is more-or-less equivalent to studying curves over Q.

But now one notices something: if we want to pursue the matter further, and make the curve y^2 = x^3 - x really look like a space, then we want to replace this equation (which is not in any concrete way a space – it is just a collection of symbols) with some actual set of points. Which set? Well, the set of rational solutions to the equation is not so good, in general, because there may not be many, or any. On the other hand, a theorem of algebra (Hilbert’s Nullstellensatz) says that the set of solutions in Q-bar is big enough to fill out the curve y^2 = x^3 - x, in any reasonable sense. And so one can use this set of solutions (equipped with its Zariski topology, and structure sheaf) as a geometric model for the curve; and now we have passed from fields and equations to actual spaces and sheaves (and so have begun to do algebraic geometry, at least in one if its many formal realizations).

Notice here that although we were studying something that had to do with a transcendence degree 1 extension of Q, the field Q-bar still played an important role. And this is one reason number theorists focus so much attention on Q-bar: because even when they want to study other, higher-dimensional, structures over Q, the set of Q-bar points is always there as a geometric model, and to understand it well requires one to understand the field Q-bar itself, and its automorphisms, arithmetic, and so on, as well as one possibly can.