### The Collatz topos

The topos-theoretic formula in the previous blogpost can be very practical in concrete calculations. Here is a fun example featuring the Collatz conjecture. To be clear, the ideas in this blog post will not help in solving the Collatz conjecture. But the example might help to gain intuition about toposes.

### Geometric morphisms to slice toposes

One of the main results about sheaves on a topological space $$X$$​​ is that they correspond to étale spaces over $$X$$​​, so to local homeomorphisms $$Y \to X$$​​. This can be turned into an equivalence of categories between the category of sheaves on $$X$$​​ and the category of local homeomorphisms $$Y \to X.$$​​ The generalization of this statement to toposes (rather than topological spaces) is well-known among topos theorists, but it took me a really long time to find this statement in the literature. It turns out the result was in SGA4… I could have seen this coming.

### Topos fundamental group of the shrinking wedge of circles

In September, I wrote about the fundamental group of a topos, but I didn’t yet give an example of how to compute this fundamental group for toposes of the form $$\mathbf{Sh}(X)$$ for $$X$$ a topological space. If $$X$$ is a CW-complex, then the fundamental group of $$\mathbf{Sh}(X)$$ agrees with the (discrete) group $$\pi_1(X)$$. On the other hand, if $$X$$ is the shrinking wedge of circles, then the fundamental group of $$\mathbf{Sh}(X)$$ is not discrete, so this is an interesting example to look at.

### Cayley graphs of torus knot groups

Knot groups are the groups that appear as fundamental groups of $$\mathbb{R}^3-K$$ where $$K \subseteq \mathbb{R}^3$$ is a knot. At the moment, I am most interested in the case where $$K$$ is the trefoil knot, because in this case the knot group is isomorphic to the braid group on 3 strands. I want to show some pictures of the Cayley graphs for knot groups, and share the Python code that I used to generate them.

### The fundamental group of a topos

This is a follow-up post of the one yesterday about the fundamental group of a monoid. There we looked at the covering spaces of the free monoid on two generators, and covering spaces of categories in general. From there it is a small(ish) step towards defining covering spaces of toposes, which in turn can be used to make sense of what the fundamental group of a topos should be.

### The fundamental group of a monoid

For a monoid $$M$$, we can look at the group $$G$$ that approximates $$M$$ in the best way. If you have a presentation of $$M$$ in terms of generators and relations, then $$G$$ has the same presentation, but as a group rather than as a monoid. In other words, $$G$$ is the group you get after formally adding inverses to the elements of $$M$$. For example, if $$M$$ is the monoid of natural numbers under addition, then $$G$$ is the group of integers.

### Riemann surfaces

I’m still trying to improve my understanding of algebraic extensions of $$\mathbb{C}(t)$$. It’s a problem that’s connected to a lot of interesting mathematical ideas, like Riemann surfaces, the étale fundamental group, dessins d’enfants and modular curves.

### Algebraic extensions and supernatural numbers

The field $$\mathbb{F}_p$$ with $$p$$ elements has a finite extension of degree $$n$$ for each natural number $$n$$. On Twitter, @syzygay1 asked if this can be extended to infinite field extensions of $$\mathbb{F}_p$$ for a generalized notion of “degree”. For the field $$\mathbb{F}_p$$, this leads to the concept of supernatural numbers, which are connected to the Arithmetic Site of Connes and Consani. It is interesting to see what happens for other fields as well, and this is what led me to write the notes below.

Let’s take the natural numbers (without zero) and order them under the division relation. So we completely forget the usual inequality $$\leq$$ between natural numbers, and we now say that $$a \leq b$$ whenever $$a$$ divides $$b$$. So $$1$$ is the smallest element and we have chains like $$1 \leq 2 \leq 4 \leq 16 \leq \dots$$. The result is called the big cell (a terminology first used by John Horton Conway). How can we best visualize this big cell? Let’s make some animated GIFs.
Take a natural number $$n \geq 2$$. Then we can create an interesting pattern as follows: we draw a dot with coordinates $$(b,c)$$ if the polynomial $$x^2+bx+c$$ is reducible modulo $$n$$. In other words, we draw a dot on $$(b,c)$$ if we can find natural numbers $$x_1$$ and $$x_2$$ such that $$b-x_1-x_2$$ and $$c-x_1 x_2$$ are both divisible by $$n$$. This pattern repeats itself vertically and horizontally.