3 ms·

Set theory has the big unexpected results of the reals being bigger than the natural numbers, the independence of the Axiom of Choice, and the undecidability of

Set theory has the big unexpected results of the reals being bigger than the natural numbers, the independence of the Axiom of Choice, and the undecidability of the Continuum Hypothesis. What are some similarly big results in category theory, for a novice?

rozgo 3y ago
Yoneda Lemma, an object is entirely determined by its relationships to other objects.
contravariant 3y ago
I'd say the Yoneda lemma should be in there. It's hard to explain without going through all the definitions but very broadly speaking it gives a precise way to characterize a object by its relations to other objects. Though mostly I consider category theory useful not for its results but because its concepts generalize well. If you can relate something to a category then most of the concepts a category has (functors, limits etc.) will have some useful meaning. This makes it easy to come up with good concepts and gives some of their properties for free, which honestly is more practically useful than some clever theorem.
- JadeNB 3y ago
  > I'd say the Yoneda lemma should be in there. It's hard to explain without going through all the definitions but very broadly speaking it gives a precise way to characterize a object by its relations to other objects. To add a bit to that, Yoneda's lemma says that you know everything about an object if you know the ways that it can be mapped to other objects. The "co-Yoneda's lemma", while often less useful in practice (in my practice, anyway), is maybe easier to understand in this intuitive way: you know everything about an object if you know the ways that other objects can be mapped to it, which I have heard phrased as something like "you can learn everything about an object by probing it with other objects."
  - solomonb 3y ago
    Coyoneda comes up with algebraic effects systems as the "Freer Monad." `Free (Coyoneda f)` gives you `Freer` which allows you to build Monads without even a `Functor` on `f`.
hackandthink 3y ago
Lawvere's fixed point theorem as a generalisation of Cantor's theorem is a big result. https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem Emily Riehl: "The author is told with distressing regularity that 'there are no theorems in category theory' ... Sadly, the majority of the theorems that are personal favorites of the author were excluded because their significance is more difficult to explain." (long list of theorems) https://math.jhu.edu/~eriehl/161/context.pdf https://math.jhu.edu/~eriehl/161/context.pdf
itigtohft 3y ago
One interesting thing is that category theory provides a way to precisely describe a Most General Unifier. It is a example of a coequalizer. This isn't a really basic and accessible result, but IMO it gives a good flavour of what category theory is suitable for: formally describing constructions we didn't previously have the tools to express precisely. https://www.sciencedirect.com/science/article/abs/pii/B9780120463701500127?via%3Dihub https://www.sciencedirect.com/science/article/abs/pii/B97801...
zmgsabst 3y ago
Algebra and geometry are the same thing: formalizing this was the motivation, originally. Semantics is the topology of your diagrams.

rozgo 3y ago

Yoneda Lemma, an object is entirely determined by its relationships to other objects.

contravariant 3y ago

I'd say the Yoneda lemma should be in there. It's hard to explain without going through all the definitions but very broadly speaking it gives a precise way to characterize a object by its relations to other objects. Though mostly I consider category theory useful not for its results but because its concepts generalize well. If you can relate something to a category then most of the concepts a category has (functors, limits etc.) will have some useful meaning. This makes it easy to come up with good concepts and gives some of their properties for free, which honestly is more practically useful than some clever theorem.

JadeNB 3y ago
> I'd say the Yoneda lemma should be in there. It's hard to explain without going through all the definitions but very broadly speaking it gives a precise way to characterize a object by its relations to other objects. To add a bit to that, Yoneda's lemma says that you know everything about an object if you know the ways that it can be mapped to other objects. The "co-Yoneda's lemma", while often less useful in practice (in my practice, anyway), is maybe easier to understand in this intuitive way: you know everything about an object if you know the ways that other objects can be mapped to it, which I have heard phrased as something like "you can learn everything about an object by probing it with other objects."
- solomonb 3y ago
  Coyoneda comes up with algebraic effects systems as the "Freer Monad." `Free (Coyoneda f)` gives you `Freer` which allows you to build Monads without even a `Functor` on `f`.

hackandthink 3y ago

Lawvere's fixed point theorem as a generalisation of Cantor's theorem is a big result. https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem https://ncatlab.org/nlab/show/Lawvere's+fixed+point+theorem Emily Riehl: "The author is told with distressing regularity that 'there are no theorems in category theory' ... Sadly, the majority of the theorems that are personal favorites of the author were excluded because their significance is more difficult to explain." (long list of theorems) https://math.jhu.edu/~eriehl/161/context.pdf https://math.jhu.edu/~eriehl/161/context.pdf

itigtohft 3y ago

One interesting thing is that category theory provides a way to precisely describe a Most General Unifier. It is a example of a coequalizer. This isn't a really basic and accessible result, but IMO it gives a good flavour of what category theory is suitable for: formally describing constructions we didn't previously have the tools to express precisely. https://www.sciencedirect.com/science/article/abs/pii/B9780120463701500127?via%3Dihub https://www.sciencedirect.com/science/article/abs/pii/B97801...

zmgsabst 3y ago

Algebra and geometry are the same thing: formalizing this was the motivation, originally. Semantics is the topology of your diagrams.