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Julius Hamilton (Jun 18 2024 at 03:41):Could someone explain what the known issues are with using set theory as a foundation for category theory?
For example, if instead of the common phrasing of the definition, “a category is a collection of objects and a collection of arrows, such that…”, we said, “a category is a set containing a set called Arrows and a set called Objects, such that…”, what are some of the immediate issues we would run into doing this? Thanks.
David Michael Roberts (Jun 18 2024 at 04:21):What then is the category of sets?
Zoltan A. Kocsis (Z.A.K.) (Jun 18 2024 at 04:34):IIRC @Mike Shulman has an article about precisely this topic! Does anyone remember the title?
David Michael Roberts (Jun 18 2024 at 04:40):@Zoltan A. Kocsis (Z.A.K.) did you mean https://arxiv.org/abs/0810.1279 "Set theory for category theory"?
Zoltan A. Kocsis (Z.A.K.) (Jun 18 2024 at 04:57):I think this is it, thanks!
Madeleine Birchfield (Jun 18 2024 at 05:08):There are also issues with certain theorems requiring the axiom of choice if categories are defined entirely in terms of sets as opposed to defined in some other foundation where groupoids or infinity-groupoids are foundational. Which might not be a problem for most people, but not everybody accepts the axiom of choice.
John Baez (Jun 18 2024 at 07:20):Julius Hamilton said:
Could someone explain what the known issues are with using set theory as a foundation for category theory?
For example, if instead of the common phrasing of the definition, “a category is a collection of objects and a collection of arrows, such that…”, we said, “a category is a set containing a set called Arrows and a set called Objects, such that…”, what are some of the immediate issues we would run into doing this? Thanks.
A category with a set of arrows and a set of objects is called a [[small category]], and you will see a lot of theorems about them. There are lots of important categories that aren't small; David mentioned the most obvious one, but there's also the category of groups, the category of monoids, the category of rings, the category of vector spaces, the category of topological spaces, etc. etc. These are [[large categories]]. If you accidentally defined "category" to mean "small category" you'd be leaving out all of these. That would be quite unfortunate.
John Baez (Jun 18 2024 at 07:32):I am deliberately not getting into questions about how we can tweak set theory to allow [[small sets]] and [[large sets]], since this shows up as a response to the problem I'm pointing out: it's a further move in the chess game, but you need to understand the first move first.