What are toposes and how are they different from sets?



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Anonymous 0 Comments

Disclaimer: I am not an expert on topos theory and I’m not sure I believe this is a subject that can be made actually ELI5.

A set is basically “a bag of stuff” where the objects have no real relation to each other except being in the same bag or not. The collection (really category, but I’m going to at least *try* to keep the language simple since this is ELI5) of all sets is a topos. But a topos can be made of things with more structure than sets, like geometric objects. Toposes have something called their “internal language” that lets you treat them sort of like sets, and translate proofs that you might easily do with sets to all toposes.

So a topos is sort of like “all sets” but generalized, so that you might have more interesting stuff going on. But they have enough structure that you can think about them like you can think about sets in a lot of ways. They allow you to do a lot of logic and geometry in a way that is more “category-theoretic” than “set-theoretic” but where some of the tools of set theory apply, and some mathematicians prefer to think this way.

There are more ways to think about toposes than “like sets” and more than one meaning of topos. It’s certainly not ELI5, but there is a good ELI-know-some-basic-category-theory intro to them that I like: [An informal introduction to topos theory, by Tom Leinster](https://arxiv.org/pdf/1012.5647.pdf)

Anonymous 0 Comments

What makes geometry “geometry”? There’s lots of ways to think about that, but one of the major themes is the notion of localization and globalization. That is, if you zoom into the surface of the earth, the geometry simplifies to something rather elementary – flat space – and so to solve problems on the surface of the earth then all you need to do is solve a bunch of these “local” problems and “glue” them together. Localization is the zooming in, and globalization is gluing them together in hopes of a solution on the whole earth (not always possible, though).

The mathematical object which contains this local-to-global structure is called a “Sheaf”. It’s complicated to define, but it gives conditions on zooming in and gives conditions on gluing things together. Most modern geometry is, in some way, done using sheaves (even if they work in the background).

What some clever people noticed, however, is that the conditions for a sheaf are pretty general and can be applied to situations that we would absolutely NOT think of as “geometry”. If we can apply a “sheaf-like thing” to some other mathematical structure, then we can basically borrow methods from geometry to solve non-geometric problems. For instance, you can apply them to the behavior of prime numbers in certain situations and basically treat prime numbers as geometric objects and do geometry to them through sheaves (this field is called “Arithmetic Geometry”). Now, this does blur the lines between what is geometry and what is not geometry, because now we can say with some certainty that anything with a sheaf is geometry now. So number theory and algebra are just iterations of geometry these days.

Topoi are a very general framework for sheaves. They are a major tool in contemporary mathematics, but also have interest with those studying foundations and logic. Basically, the Category of Sets is a very trivial topos, and so certain more complicated topoi can be thought of as an enrichment of the Category of Sets – if that’s what you’re into. When people talk about topoi in the context of foundations or logic, they mean this “geometric enrichment” of set theory through the use of sheaves. There are minor difference here and there, but the idea is the same.

There is this idea that once math is done, it’s done. The calculus you learn in college is the same calculus that they had in 1840 when Weierstrass wrote down the full definition of a limit (or, one of his students did). But math evolves and goes through vibe shifts all the time. The history of math for the last century has, more or less, been dominated by the philosophy of sheaves. Almost everything can be formulated into sheaves and sheaves give powerful tools to solve previously unsolvable problems. Such changes are largely invisible to those outside of math, but I honestly think that sheaves are one of the most significant intellectual developments in human history.