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in Chat

I'm a postdoc working in topology and set theory. I've been interested in learning about categorical foundations just to see what the issues and techniques are there. But there aren't many people around me who work in this area, and there even seems to be some tension here with set theorists (at least on discussions in MathOverflow).

Category theory background: I took some algebra/algebraic geometry classes that started with category theory and always found the basic definitions very intuitive, but I haven't been exposed to it on a deep level.

## Comments

Hello, William! What aspects of set theory and topology are you working on? Where are you a doing your postdoc?

There's a very productive interaction between set theory and category theory, which is unfortunately held back at times by silly disputes between people who like one more than the other. One thing I like is how some large cardinal axioms can be cleanly expressed as statements about category theory.

For example, Vopěnka's principle says that asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. It's equivalent to some statements that are easier for me to understand, like this: For every proper class of simple directed graphs, there are two members of the class with a graph homomorphism between them. But it's also equivalent to some very nice statements in category theory: for example, every discrete full subcategory of a locally presentable category is small.

I don't understand this stuff very well at all, so don't ask me questions about it - and don't be intimidated! My own work usually concerns finite things, no scary infinities. I just think it's nice to see fancy aspects of set theory connecting to category theory in this way.

A more practical thing, in many ways, is the elementary theory of the category of sets.

`Hello, William! What aspects of set theory and topology are you working on? Where are you a doing your postdoc? There's a very productive interaction between set theory and category theory, which is unfortunately held back at times by silly disputes between people who like one more than the other. One thing I like is how some large cardinal axioms can be cleanly expressed as statements about category theory. For example, [Vopěnka's principle](https://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle) says that asserts that for every proper class of binary relations (each with set-sized domain), there is one elementarily embeddable into another. It's equivalent to some statements that are easier for me to understand, like this: For every proper class of simple directed graphs, there are two members of the class with a graph homomorphism between them. But it's also equivalent to some [very nice statements in category theory](https://ncatlab.org/nlab/show/Vop%C4%9Bnka%27s+principle): for example, every discrete full subcategory of a locally presentable category is small. I don't understand this stuff very well at all, so don't ask me questions about it - and don't be intimidated! My own work usually concerns finite things, no scary infinities. I just think it's nice to see fancy aspects of set theory connecting to category theory in this way. A more practical thing, in many ways, is the [elementary theory of the category of sets](http://www.maths.ed.ac.uk/~tl/edinburgh_yrm/edinburgh_yrm_talk.pdf).`