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What does it mean, that it's replacing set theory? Is it more fundamental than set theory, in the sense that the latter can be defined in terms of it? Or is it

What does it mean, that it's replacing set theory? Is it more fundamental than set theory, in the sense that the latter can be defined in terms of it? Or is it different from set theory entirely, meaning different math comes from each?

Twisol 4y ago
Category theory might be easier to think about as a different way to think about the same objects we've always studied in mathematics. Since it's a formal theory (just like set theory), we can also study category theory mathematically. In fact, we can study set theory categorically and we can study categories set-theoretically -- they aren't really at odds with each other. If you wanted to study set theory from a categorical perspective, you would likely be interested in topos theory, which captures essential properties of set theory and generalizes them to a variety of interesting settings. There's also a really good (and IMHO approachable) paper by Tom Leinster [1] on how to see set theory from a categorical perspective, without going all the way out to toposes. I think it's a good way to understand what category theorists try to emphasize. If you wanted to study categories from a set-theoretic perspective, you're in luck -- that's probably the most common approach to learning category theory in the first place! Studying categories with categories is also possible, but you already need to be fairly comfortable with categories in order to model them internally. (Studying set theory with set theory is much the same -- models of ZFC can get wild.) (You can also learn category theory from first principles, without thinking in terms of sets. This isn't really any different from set theory, either -- set theory is often peoples' first introductions to abstract mathematics at the undergraduate level, so there isn't really much choice in the matter. For my money, though, I think category theory from first principles is a little less weird.) [1]: https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543
i_k_k 4y ago
I never understood this either... Category theory works with collections, which can be modeled as sets -- or if you want to deal with categories of sets and other "large" categories -- as proper classes. And you can use a set theory that incorporates classes, like Morse-Kelley, to model this. IOW, it seems like category theory can be modeled in set theory. On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". Set theory also sheds a lot of light on what feel like foundational issues: sizes of sets, independence of axioms, etc.; I haven't seen something similar out of category theory. Love for a category theory partisan to chime in with more here -- I'm definitely not an expert.
- Twisol 4y ago
  > On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". I'm no expert, but I do want to point out that you'll never see a set of axioms for set theory that doesn't also presuppose some realm from which sets themselves are drawn. The question is not whether sets exist; it's whether the collection of them can be represented within the theory. It is well-known (in certain circles...) that the realm of all sets cannot itself be described by a set. In the same way, category theory may assume the existence of collections of objects and arrows, but these collections are not themselves represented within category theory. The category of all categories encounters exactly the same size issues as does the set of all sets. There are different ways around these size issues. I think a lot of people go with something like Russell's stratified hierarchy of universes -- but this is also done in set theory, like with the alternative to ZFC based on sets and classes. (But what is the class of all classes!? You just keep building bigger universes.)
  - lmm 4y ago
    > I'm no expert, but I do want to point out that you'll never see a set of axioms for set theory that doesn't also presuppose some realm from which sets themselves are drawn. Huh? Of course you do; ZFC itself is such a set of axioms. There's no particular requirement on objects that can be members of sets (at least, nothing really goes wrong if they're not all sets), so the theory doesn't depend on anything else, in a very intuitive sense.
    - Twisol 4y ago
      To get a little bit technical, the axioms of ZFC, as a "first-order theory", make use of universal and existential quantifiers. These quantifiers have to quantify over some universe, whose constituents we call "sets". Take the axiom of the empty set as an example: exists x. forall y. y not in x The meaning of the "exists" quantifier is that it runs over the universe of entities; the formula it quantifies over must be true when instantiated on at least one such entity. The meaning of the "forall" quantifier is similar, except it must be true for every such instantiation. Notably, since the universe of all sets is not a set on pain of paradox, the universe itself cannot be a set. It is a primitive "sort", part of the logical signature of the theory. These quantifiers only "work" when there's a universe of entities to run over. This universe is not internal to set theory as an axiomatic system; the whole point of "non-standard models" is that we're looking for universes where the axioms of set theory hold, and yet are not what we normally think of when we reason using those axioms (e.g. "large cardinal" models). Category theory can also be framed as a first-order theory. In its most common incarnation, it's a two-sorted first-order theory, with separate universes for objects and for arrows. This isn't essential; we can actually dispense with objects and work only with arrows. (Objects are then encoded as their identity arrows.) But it's a first-order theory either way; all the axioms of a category can be given with quantified formulas. Group theory is also a first-order theory. When you specify a particular group, you give the universe over which its axioms quantify, and then prove(!) that those axioms hold on that universe. If you interpret group theory internal to set theory, then your universe will be a set; but the axioms don't care as long as quantification can be interpreted appropriately.
layer8 4y ago
Category theory is not replacing set theory. It's also not the case that entirely different math comes from each. Both can be defined in terms of each other to a large degree. It's probably better to think of them complementing each other. You might find the following thread interesting to read: https://mathoverflow.net/questions/360578/category-theory-and-set-theory-just-a-different-language-or-different-foundati https://mathoverflow.net/questions/360578/category-theory-an...

Twisol 4y ago

Category theory might be easier to think about as a different way to think about the same objects we've always studied in mathematics. Since it's a formal theory (just like set theory), we can also study category theory mathematically. In fact, we can study set theory categorically and we can study categories set-theoretically -- they aren't really at odds with each other. If you wanted to study set theory from a categorical perspective, you would likely be interested in topos theory, which captures essential properties of set theory and generalizes them to a variety of interesting settings. There's also a really good (and IMHO approachable) paper by Tom Leinster [1] on how to see set theory from a categorical perspective, without going all the way out to toposes. I think it's a good way to understand what category theorists try to emphasize. If you wanted to study categories from a set-theoretic perspective, you're in luck -- that's probably the most common approach to learning category theory in the first place! Studying categories with categories is also possible, but you already need to be fairly comfortable with categories in order to model them internally. (Studying set theory with set theory is much the same -- models of ZFC can get wild.) (You can also learn category theory from first principles, without thinking in terms of sets. This isn't really any different from set theory, either -- set theory is often peoples' first introductions to abstract mathematics at the undergraduate level, so there isn't really much choice in the matter. For my money, though, I think category theory from first principles is a little less weird.) [1]: https://arxiv.org/abs/1212.6543 https://arxiv.org/abs/1212.6543

i_k_k 4y ago

I never understood this either... Category theory works with collections, which can be modeled as sets -- or if you want to deal with categories of sets and other "large" categories -- as proper classes. And you can use a set theory that incorporates classes, like Morse-Kelley, to model this. IOW, it seems like category theory can be modeled in set theory. On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". Set theory also sheds a lot of light on what feel like foundational issues: sizes of sets, independence of axioms, etc.; I haven't seen something similar out of category theory. Love for a category theory partisan to chime in with more here -- I'm definitely not an expert.

Twisol 4y ago
> On the other hand, I've not seen a rigorous set of axioms for category theory that didn't presuppose a notion of set or category or "collection". I'm no expert, but I do want to point out that you'll never see a set of axioms for set theory that doesn't also presuppose some realm from which sets themselves are drawn. The question is not whether sets exist; it's whether the collection of them can be represented within the theory. It is well-known (in certain circles...) that the realm of all sets cannot itself be described by a set. In the same way, category theory may assume the existence of collections of objects and arrows, but these collections are not themselves represented within category theory. The category of all categories encounters exactly the same size issues as does the set of all sets. There are different ways around these size issues. I think a lot of people go with something like Russell's stratified hierarchy of universes -- but this is also done in set theory, like with the alternative to ZFC based on sets and classes. (But what is the class of all classes!? You just keep building bigger universes.)
- lmm 4y ago
  > I'm no expert, but I do want to point out that you'll never see a set of axioms for set theory that doesn't also presuppose some realm from which sets themselves are drawn. Huh? Of course you do; ZFC itself is such a set of axioms. There's no particular requirement on objects that can be members of sets (at least, nothing really goes wrong if they're not all sets), so the theory doesn't depend on anything else, in a very intuitive sense.
  - Twisol 4y ago
    To get a little bit technical, the axioms of ZFC, as a "first-order theory", make use of universal and existential quantifiers. These quantifiers have to quantify over some universe, whose constituents we call "sets". Take the axiom of the empty set as an example: exists x. forall y. y not in x The meaning of the "exists" quantifier is that it runs over the universe of entities; the formula it quantifies over must be true when instantiated on at least one such entity. The meaning of the "forall" quantifier is similar, except it must be true for every such instantiation. Notably, since the universe of all sets is not a set on pain of paradox, the universe itself cannot be a set. It is a primitive "sort", part of the logical signature of the theory. These quantifiers only "work" when there's a universe of entities to run over. This universe is not internal to set theory as an axiomatic system; the whole point of "non-standard models" is that we're looking for universes where the axioms of set theory hold, and yet are not what we normally think of when we reason using those axioms (e.g. "large cardinal" models). Category theory can also be framed as a first-order theory. In its most common incarnation, it's a two-sorted first-order theory, with separate universes for objects and for arrows. This isn't essential; we can actually dispense with objects and work only with arrows. (Objects are then encoded as their identity arrows.) But it's a first-order theory either way; all the axioms of a category can be given with quantified formulas. Group theory is also a first-order theory. When you specify a particular group, you give the universe over which its axioms quantify, and then prove(!) that those axioms hold on that universe. If you interpret group theory internal to set theory, then your universe will be a set; but the axioms don't care as long as quantification can be interpreted appropriately.

layer8 4y ago

Category theory is not replacing set theory. It's also not the case that entirely different math comes from each. Both can be defined in terms of each other to a large degree. It's probably better to think of them complementing each other. You might find the following thread interesting to read: https://mathoverflow.net/questions/360578/category-theory-and-set-theory-just-a-different-language-or-different-foundati https://mathoverflow.net/questions/360578/category-theory-an...