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Life in Johnstone's Topological Topos 3 -- Bonus Axioms

In the first post of the series, we talked about what the topological topos is, and how we can think about its objects (and, importantly, how we can relate computations in the topos $\mathcal{T}$ to computations with topological spaces in “the real world”). In part two, we talked about algebraic structures, and how (for example) groups in $\mathcal{T}$ are related to topological groups.

Featured Posts

• Life in Johnstone's Topological Topos 1 -- Fundamentals

  I’ve been thinking a lot about the internal logic of topoi again, and I want to have more examples of topoi that I understand well enough to externalize some statements. There’s more to life than just a localic $\mathsf{Sh}(B)$, and since I’m starting to feel like I understand that example pretty well, it’s time to push myself to understand other important examples too!

• Internal Group Actions as Enriched Functors

  Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural question about how we might study topological group actions via the functorial approach beloved by category theorists. The usual story is to treat a group $G$ as a one-object category $\mathsf{B}G$. Then an action $G \curvearrowright X$ is the same data as a functor $\mathsf{B}G \to \mathsf{Set}$ sending the unique object of $\mathsf{B}G$ to $X$. Is there some version of this story that works for topological groups and continuous group actions?

• A truly incredible fact about the number 37

  So I was on math stackexchange the other day, and I saw a cute post looking for a book which lists, for many many integers, facts that Ramanujan could have told Hardy if he’d taken a cab other than 1729. A few days ago OP answered their own question, saying that the book in question was Those Fascinating Numbers by Jean-Marie De Koninck. I decided to take a glance through it to see what kinds of facts lie inside (and also to see just how many integers are covered!). Not only was I overwhelmed by the number of integers and the number of facts about them, the preface already includes one of the single wildest facts I’ve ever heard, and I have to talk about it here! Here’s a direct quote from the preface:

• Externalizing Some Simple Topos Statements

  Hey all! It’s been a minute. I’ve been super busy with the UC strike and honestly I haven’t done math in any serious capacity for almost the past month. It’s been a lot of hard work trying to get fair contracts out of the UC, but I had a lot of travel plans this December to see my friends, so I’ve taken a step back from the picket line until January. Right now I’m in Chicago, so here’s an obligatory bean pic:

• Monoidal Monoidoidoids

  So I was on the nlab the other day, and I saw a fantastic joke: A 2-category is “just” a monoidal monoidoidoid! Here’s a screenshot in case the nlab page for 2-categories changes someday:

• Talk -- Let's Solve a Simple Analysis Problem. Together.

  Last Friday I gave a talk in GSS where I tried to give a super concrete application of topos theory to “mainstream” mathematics. The idea was to solve a simple analysis problem, streamlining the argument by using the internal logic of a sheaf topos. The good news is that I think I was quite successful in making my point that “ordinary mathematicians” should care about topos theory and constructive logic. The bad news is that the last 10 minutes of my talk were false… It didn’t end up mattering, but I was still pretty torn up about it. Anyways, in this post I’ll give an overview of the talk, which should double as a nice description of how to actuallly use topos theory to solve problems. I’m planning to start a new series soon where I go over this proof in more detail, explaining the relevant aspects of topos theory along the way!

• Slaughtering Competition Problems with Quantifier Elimination

  Anytime I see questions on mse that ask something “simple”, I feel a powerful urge to chime in with “a computer can do this for you!”. Obviously if you’re a researching mathematician you shouldn’t waste your time with something a computer can do for you, but when you’re still learning techniques (or, as is frequently the case on mse, solving homework problems), it’s not a particularly useful comment (so I usually abstain). The urge is particularly powerful when it comes to the contrived inequalities that show up in a lot of competition math, and today I saw a question that really made me want to say something about this! I still feel like it would be a bit inappropriate for mse, but thankfully I have a blog where I can talk about whatever I please :P So today, let’s see how to hit these problems with the proverbial nuke that is quantifier elimination!

• Topological Categories -- A Unifying Framework

  I think there is an obvious analogy that most mathematicians notice, perhaps subconsciously, when learning about topological spaces and measure spaces (often within a year or two of each other). The definitions look similar, as do the associated maps (continuous maps pull back open sets to open sets, and measurable maps pull back measurable sets to measurable sets). Seeing this, it’s reasonable to ask if measure spaces and topological spaces share any similarities. More abstractly, one can ask whether their categories share any similarities. The answer turns out to be a resounding yes, and the study of these Topological Categories will be the subject of today’s post!

• Dense Pythagorean Triples -- A Cute Problem in Arithmetic Geometry

  I follow a lot of other math blogs, and it’s always fun reading about what various people are thinking about. Particularly fun for me is when blogs pose problems to their readers, and the other day Shiva Kintali posted a really cute problem: For any rational numbers $\alpha \lt \beta \in (0,1)$, can we always find a pythagorean triple $a^2 + b^2 = c^2$ so that $\alpha \lt \frac{a}{b} \lt \beta$?

• A Nonarithmetic Example of a Noneuclidean Principal Ideal Domain

  This blog post changed dramatically over the course of writing it… I’m keeping the title, because I like the alliteration of “nonarithmetic noneuclidean”, but by the end I actually understood the arithmetic example better too. Perhaps a better title would be “In which algebraic geometry explains noneuclidean PIDs in a way that finally makes sense to me”.

• Iteration Asymptotics

  I really like recurrences, and the kind of asymptotic analysis that shows up in combinatorics and computer science. I think I’m drawn to it because it melds something I enjoy (combinatorics and computer science) with something I historically struggle with (analysis).

• Two Sage Visuals

  I’m in a reading group with Elliott Vest and Jacob Garcia (supervised by Matt Durham) where we’re talking about CAT(0) Cube Complexes. We’re reading a set of lecture notes by sageev (pdf here, for the interested) and we came across a fairly simple problem that we wanted to draw. In a completely different vein, Russell Phelan asked a fun topological question in the UCR math discord, and to solve it I ended up needing to draw something else. I figured I would write up a quick post about both visualizations, since these things can be a bit tricky to get right.

• Cohomology Intuitively

  So I was on mse the other day…

• Linearly Ordered Groups and CH

  Earlier today Jonathan Alcaraz gave a GSS talk about Linearly Ordered (LO) Groups, which are a fun topic with connections to dynamics, topology, geometric group theory, etc. This reminded me of a problem I told myself to think about a while ago, and so I decided to finally do that. After a bit of thought, a friend from CMU (Pedro Marun) and I were able to figure it out. This post is going to be somewhat more meandering than usual (if you can imagine such a thing), because I want to showcase what the flow of thoughts was in solving the problem. At the end I’ll clean things up and write them linearly.