# What are toposes and how are they different from sets?

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[https://en.wikipedia.org/wiki/Topos](https://en.wikipedia.org/wiki/Topos)

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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)