# Small ideas on not-quite-covering maps

If you have done any algebraic geometry, then you will have come across descriptions of “covering families” of various types, usually described as a collection of maps with a) common codomain, b) some nice property, such that c) they are jointly surjective. For instance, an Zariski/étale/smooth/flat covering family of a scheme $X$ is a collection of open embedding/étale/smooth/flat maps $U_\alpha \to X$ such that $\coprod_\alpha U_\alpha \to X$ is an epimorphism. (one might need more adjectives over mere flatness, but these can be safely added with no change of anything below.) One can play similar games in other categories; there is a notion of cover of manifolds that is a surjective submersion (or, in principle, a jointly surjective family of submersions), or a cover of a topological space that is an open surjection. That surjectivity is here is good and right: this makes the covers/covering families into subcanonical pretopologies, in that in the case of the single covering maps, these are regular epimorphisms, and really do capture the notion of “being a cover” (in the covering family version, an analogous property holds).

But what is common to the examples above (and many others) is that the other property also has nice features. Submersions, open maps, open embeddings, étale maps, smooth maps etc all contain the isomorphisms, are closed under composition, and are closed under pullback. As such these classes of maps, not assuming surjectivity, already form Grothendieck pretopologies, though they are far from being subcanonical. Further, the categories of schemes, manifolds and topological spaces exhibit a property called extensivity. As a result, coproducts behave as you expect disjoint unions to, rather than, say, direct sums of vector spaces (which are also coproducts). Moreover, the coproduct inclusions form a covering family in each of the cases given (so we have a superextensive pretopology). This implies that one can turn each of these pretopologies into a pretopology where every covering family consists of a single map (a singleton pretopology). This process preserves the property of being subcanonical, and so I will assume that we are working with a singleton subcanonical pretopology.

Another feature that these examples display is that given, for instance, a submersion $X \to N$, then given any surjective submersion $M \to N$, the resulting fold  map $X \sqcup M \to N$ is a surjective submersion. The same is true for open maps, for open embeddings, for étale maps and so on. There is a recent PhD thesis by Giorgi Arabidze (not this Giorgi Arabidze!), that deals with internal groupoids in categories equipped with an abstraction of this type of structure, though I have a slightly different take. He starts from an arbitrary singleton pretopology (though calls it a “stronger pretopology”), whose maps are called partial covers, then singles out the subclass of partial covers that are also regular epimorphisms, and calls those covers. The covers then form a subcanonical singleton pretopology. I’m interested in starting from the latter, and seeing what sort of larger classes of maps I can take that have the fold map property above.

As an example, whenever a subcanonical superextensive pretopology is given by specifying that a covering family is a collection of jointly-surjective $X$-maps, for $X$ a class of maps that form a pretopology on their own, then this is the sort of setup I’d like to abstract, but merely starting from a subcanonical singleton pretopology. So here is a first attempt at this. Nothing deep, just wanting to see if it seems like something that’s out there (and the name is totally provisional).

Definition: Let $J$ be a subcanonical singleton pretopology on an extensive category $S$. Let $K \supset J$ be another singleton pretopology such that

1. Given any $U \to X$ in $K$ and $V\to X$ in $J$, the fold map $U+V\to X$ is in $J$
2. $K$ is local for the extensive pretopology.

We then call $K$ a class of precovers for $J$.

By “local for the extensive topology” I mean that if I have a map in $K$ with codomain a coproduct, if its corestriction to each summand is in $K$, then it is itself in $K$. This implies that the coproduct $f+g\colon U+V\to X+Y$ of maps in $K$ is again in $K$. What I am interested in is the largest possible class of precovers for a given pretopology as in the definition. As noted in the title, these maps are not quite covers, but can be ‘completed’ to a cover. Another way to think about this might be something like finding a class of maps in which $J$ is a kind of ‘ideal’ for the operation of forming the fold map in a suitable slice category.

As an example that is somewhat contrasting to the above more geometric ones, consider an extensive category $S$ with all finite limits (a ‘lextensive’ category; a topos is an example), a subcanonical singleton pretopology $J$, and consider the subcanonical singleton pretopology given by maps that admit $J$-local sections. Then the class of all maps in $S$ form a class of precovers. For an more specific example of interest, take a pretopos and the singleton pretopology $\mathrm{Reg}$ consisting of the regular epimorphisms (this, together with the extensive pretopology generate the coherent pretopology). What are the maps that admit $\mathrm{Reg}$-local sections? And then what is the largest class of precovers, is it all maps again?

# AustMS 2019 talks

Just so I have these somewhere, these are the notes/slides for the two talks I gave at the annual Australian Mathematical Society talk last week. The first is a revision of this talk.

# Terry Tao’s first paper: “Perfect numbers”

Terry Tao has just been awarded the inaugural Riemann prize, and as a result I discovered he had his first mathematics paper published at age 8, in a (now defunct) journal for school mathematics in my home state of South Australia. Since this rare item only appears available reproduced as an appendix in a scanned pdf of a 1984 article in an education journal, I thought I’d re-typeset it. So here it is:

Terence Tao, Perfect numbers, Trigon (School Mathematics Journal of the Mathematical Association of South Australia) 21 (3), Nov. 1983, p. 7–8. (pdf)

Note that this appeared 13 years before his earliest listed paper in Math Reviews/MathSciNet.

Edit: I made a GitHub repository for the paper, the LaTeX source, and the code (working, after minor edits) from it. I passed it on to Tao already.

# Diagonal arguments: once more with feeling

I’ve been thinking more about the diagonal argument this week, and realised my version for magmoidal cats with diagonals was insufficient to cover a massively important example: the category of partial recursive functions! This is because every partial endomorphism $\mathbb{N} \to \mathbb{N}$ has a fixed global point, namely the undefined point: the partial function $1 \to \mathbb{N}$ with empty domain. The same problem holds for arbitrary generalised points $X \to \mathbb{N}$. So the diagonal argument can’t get started. Any general diagonal argument should be able to deal with the special case of partial recursive functions without special tweaks to deal with such behaviour. So while my magmoidal diagonal argument is valid, it needs more work to apply where one has partial functions.

# The only curve with a unique focus is the parabola

Recently, an article with a breathless headline (since revised) was published on the ABC website about how a year 12 student in regional Victoria, Mubasshir Murshed, published a proof of the above result in an academic journal. I think this a good achievement and to be commended, but the way it is framed is…problematic.  The original headline, “Teenager’s parabola equation blows away maths world” could have done with some input from mathematicians (I understand how news works, the journalist probably had little choice in the matter). The quote from the editor of the Australian Mathematics Education Journal also seems to me to lack perspective. Publishing in an education journal is not the same as publishing in a mathematics journal, which requires something genuinely previously undiscovered.

I’m all for furthering the idea that everyone can discover interesting mathematics themselves, even if it was already known. That’s part of the fun in mathematics, figuring out how to understand existing work in a new way, or even the joy of realising you discovered a piece of ‘real’ mathematics that someone else had thought of. I definitely had fun deconstructing the proof and rewriting it (below!) in a way I felt more comfortable with. But heralding someone who does this as a “17-year-old mathematics genius” is, I feel, counterproductive to the promotion of mathematics more broadly. It can definitely prompt people to put themselves in the “I’m not a genius, I can’t do mathematics” box, when this is just not true. My colleague David Butler engages with people of all kinds of backgrounds with public mathematical play, and they often are doing little pieces of mathematical discovery without even realising it. Praising up doing mathematics as a work of “genius” alienates people.

It’s easy to praise such an achievement as Murshed’s, but the article missed an opportunity to highlight the way in which discovery in mathematics is cheap: you don’t need any special equipment! With access to the internet, and pen and paper there is little limit to what kinds of theoretical mathematics you can do. The idea that problems such as the result in the title are even amenable to proof is genuinely important. That abstract-looking mathematics can have a real-world impact (parabolic reflector design!) is important to know for non-mathematical policy-makers at the all levels, but also for people genuinely if there is to be an appreciation of mathematics in today’s society. There is also the message that mathematics is not ever truly finished, and that there can be more than one way to approach a theorem and its proof (see below).

With the grumpiness out the way (and I’m happy to discuss the pros and cons), I want to give a variation on the theorem and proof, assuming less than Murshed’s version. I am happy to take as a black box one of the equations he derives, because that is done by elementary geometry.

Theorem: A plane curve given by the graph of a differentiable function and with a unique focus is a parabola.

# Third solution to writing 3 as a sum of three third powers!

Andrew Booker and Andrew Sutherland have found, using the the Charity Engine distributed computing platform, a third solution in integers to the equation $x^3 + y^3 + z^3 = 3$, so we now know each of

• 13 + 13 + 13

• 43 + 43 + (-5)3

• 5699368212219623807203 + (-569936821113563493509)3 + (-472715493453327032)3 (verify!)

is equal to 3. It is conjectured that there are infinitely many integer solutions, but we seem to be nowhere near that.