Crap, you're right; only finite unions and intersections are inherited from the power set. Completeness is another matter... thanks for pointing this out! *Edit:* unfortunately, @yuvalpaz3752 showed also that A cannot be complete at all, meaning I have only provided an example of a (not complete) Boolean algebra with no atoms. For a *complete* example that's similar to this, you can take the algebra of Lebesgue measurable subsets of R, where two subsets are considered essentially the same if they differ by a set of Lebesgue measure zero. The argument follows the same general idea, but this example is actually complete as a Boolean algebra. Sorry everyone!
@@SheafificationOfG I'm not sure exactly how you would define them, but I have a hunch that you'd be fighting the axiom of choice in favour of the axiom of determinacy to get everything you're looking for.
*Edit:* the content written below is nonsense, and is simply false (thanks @yuvalpaz3752 for calling me out!). Turns out the algebra A is not complete. :( Sorry everyone! Ok, now that I have a bit of time, here's the construction (no need for AoC)! I'll describe joins, meets are analogous [dual, if you will]. For each element x of A, define its "ideal" [x] to be the set of all subsets of N that are essentially the same as some subset of x. You can identify x with its ideal (x is essentially the same as every maximal element of its ideal), and conversely, any subset of P(N) that is closed under inclusions and essential equality is the ideal of some (essentially unique) element of A). Now, this may be a confusing set of words, but to compute the join of an infinite family (x_i)_i of elements x_i of A (indexed by some infinite set I), do the following. 1. For every finite subset J of I, compute the union u_J of the x_j for j in J (this can be done set-theoretically). 2. For each u_J, compute its ideal [u_J]. 3. Since each ideal [u_J] is a subset of P(N), take the union of these [u_J] over all finite subsets J of I. This is a new subset U of P(N). 4. U is closed under inclusions and essential equality, and is therefore the ideal of some (essentially unique) element of A. This element is precisely the join of the x_i. [In fancy categorical nonsense, the above strategy is how to compute filtered colimits in A viewed as a category, and since we can compute finite coproducts (unions), arbitrary unions can be written as a filtered colimit of finite unions.]
@@SheafificationOfG your definition is problematic, in step 4 you took union of "ideal"s and claim this to be an "ideal". Look at x_i={p_i^n | n in N} for p_i the i-th prime. Take the union of [u_J] for u_J as you defined, call it U. If U=[A] then A contains all but finitely many elements from x_i for each i, let z_i the first power of p_i that is in A and define B to be {x in A | x is not equal to z_i for any i}. A simple argument shows that each u_J is in [B], in particular U is a subset of [B], but A is not in [B] by definition. Edit: a similar argument shows that P(N)/fin is not complete under any sensible order
@yuvalpaz3752 crap you're absolutely right. P(N)/fin is not complete at all... my bad. (Though in defense of the term "ideal", these are ideals in the order-theoretic sense... my mistake was that U is not a principal ideal, as shown by your argument.)
love love LOVE this channel. So glad I found this while I'm still in the knows-enough-about-category-theory-to-follow-the-video-but-not-enough-to-already-know-everything-in-the-video stage. Awesome.
I am constantly amazed by how well you blend category theory and internet memery; I keep dipping my toes into category theory and suddenly finding myself over my head, but you're one of the best life rafts (to really drag out this analogy) that I've found. Thank you! Minor nitpickery: at 2:53 I think you have a couple of the composition formulae wrong. sr needs to be s, not r (both are arrows from top-middle to top-right) and vs and v are likewise between different objects of C; I'm pretty sure that has to be vs=u almost definitionally since with {0} a terminal object every arrow from top-middle to bottom is just u.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda.
This takes me back to mathematical logic in my first year of uni. It's really hard to grasp how, even for very philosophically abstract considerations you can run into fun math problems (like the combinatorial argument you showed at the beginning for the difficulty of finding which functions can be preimages).
1:01 the set theorists are the ones who will hunt you down. in mode theory you pretend there is a university (usually called a monster) which is a set all of the time.
In model theory, a set in one model of set theory may not be a set in another. And also, the elements of a set aren't property of the set itself but of the set membership relation. So in model theory, there's no such thing as small or large collections, only elements and non-elements of a model. Actually there is if one refers to the meta-language (e.g. elementary class), but that's more like the concern of set theorists.
@@Noname-67 no, i didn’t mean the model theory of set theory. i meant, you fix a cardinal κ that’s large enough for your whatever you are trying to do, you fix a “monster” model M of your theory that is κ-saturated and you pretend M is the universe. you wouldn’t do this with ZFC because it has no good model theoretic properties (you can not even find good ways of completing it), but you would with other theories (like algebraically closed fields or something).
i'm very early on in this but i have some conflicted feeling, namely about considering the defining property of a set as being an object in the category of sets *and functions as morphisms*. maybe it's too pedantic but from the perspective of sets and sets alone, it's not apparent that functions are singled out as special operations... relations seem just as fundamental. and Rel is self-dual!
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda.
So glad I found your channel Sheafification of G! This was something I actually thought about briefly but never bothered to try; seeing the answer put together in such an entertaining way was a real joy to watch! You got my sub!
I probably could understand this, if I turned on my brain. Instead I'll just enjoy some fancy animations and go "yup, makes sense" at the end. It's a nice video, and somehow gives me the vibes of a tech DIY youtuber showing off a new toy they made, except it's a mathematician creating new branches of math on a napkin. I have a hunch that a lot of effort went into discovering every line of equations that was shown. I wish I had a solid enough grasp of the field to do the same. Seems like magic. Rock on, science man :)
I've never tried to understand boolean algebras as I thought they were not very fun. That is until I see this video. I love the way you presentet this, especially that the term boolean algebra only appeared halfway into the video so I didn't have time to run away lol
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
"You might have noticed that, in some bold and brash way, every power set contains its original set: the elements of the original set each define a unique one-element subset." I got flashbacks to learning about NFU, where a power set can be smaller than the original set because the injection x ↦ {x} can't be created
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
As an undergrad maths student, just wanted to say I absolutely love your videos - perfect pacing, topics, humour, random shitty osu! references - honestly, perfect
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
It's after midnight on a Friday... I guess Saturday now, and I'm watching this after drinking whiskey for the past few hours. Not entirely sure why this one came up, but it made sense and is well done. Thanks for making this and now it is time to sleep!
Opposite is a (contravariant) functor, and the opposite of a functor is the same functor but on the opposite categories. Since Opposite is a functor mapping a category to its opposite, co-Opposite is a functor mapping the opposite a category to said category. This is simply the same as Opposite. In other words, the opposite of Opposite is simply the Opposite. Which means that the co-opposite of a set is simply "coset".
At first I was confused by this, since I’d assume the co-opposite of set would be set itself. Here’s the analogy that worked for me if anyone else is confused. Think of involutions (functions that are their own inverse), like “negative”. The function “negate” sends positive numbers to negative ones and vice-versa (with 0 staying where it is). The opposite of this sends negative numbers to positive ones and vice-versa… which is just the same as negating. This is why negative-negative makes positive. Notably, finding the inverse of a function reverses _the behaviour of the function_ but _doesn’t affect its input._ Similarly, applying “co” to a functor reverses its behaviour, not its input. So if a functor is its own cofunctor, its behaviour is unchanged. The “opposite” function can be thought of like “negate”, since opposite-opposite is just no change. Therefore co-opposite is just opposite, and applying it to set gives back the opposite of set, analogously to how -^(-1) x = -x and not +x. Neat!
This is very cool. A simpler (though probably less interesting) answer is that the opposite of Set is the category whose objects are sets and whose morphisms are injective and surjective relations.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
13:08 I don't get why (Assuming excluded middle) the video doesn't talk about complements as it is a part of the definition of boolean algebras. for inverse images g, g(A\B) = g(A)\g(B) and X\(A U B) = X\A intersect X\B. Edit: I think I get it, kinda got swept under the rug but since we are only considering the complete lattice of power sets under set inclusion, well that particular example is orthocomplemented. Though not all complete lattices are orthocomplemented. What is kinda interesting here is that homomorphisms of completed lattice can preserve orthocomplmention somehow. probably something like the homomorphism preservation theorem?
Ayo there is a spooky amount of overlap between my coordinate function-space, and this coset stuff you made. 😳 I'm currently working on it with professors and it will likely be published to academia in the coming weeks. I'll be sure to shout you out, Given you managed to scrape up just some of this coordinate function-spaces properties on your own. I'll also Email send you the paper when i publish it to keep you up to date. 😁👍
You can also get some notion of coset from linear type theory. There you can get back to usual type theory via !. In some contexts ! Is the cofree (cocommutative?) comonoid functor, which implies that regulat types/sets are those linear types which are (cocommutative) comonoids. Dually cosets would be (commutative) monoids, but over ⅋ (par) not the usual product. But because linear logic is self dual they just correspond to the negaton of regular sets, which is a bit boring.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
2:27 you say “some people, like analysts…” I got my PhD in mathematical analysis, I can assure you we call ourselves “Analysists” not “Analysts” as the latter is a different job entirely.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Learning that I've spent too much time studying mathematics because I can guess what the answer is going to be after a couple of minutes is a fun way to start my morning
this was such a cool video about something i've been thinking about myself! specifically i was thinking about an Ind-dual to condensed sets. if you Ind-complete (rather than Pro-complete) the category of finite sets you get countable sets, and product preserving presheaves on these form the so called "bornological topos". i feel like an algebraic method could be used on these similar to condensed sets. this video is inspiring me to give this another go
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Very nice vid thank you :D I'm new to category theory and while I understand the yoneda lemma superfically I don't have good intuition for it. Could someone elaborate on the statement G makes at 1:52, about how all objects are set objects? I don't see how this follows from the yoneda lemma but it seems like it could be a helpful fact to think about general categories.
Glad you liked the vid! Essentially, this is an interpretation of how the Yoneda embedding is fully faithful, or that objects are uniquely determined up to isomorphism by their representable presheaf. Basically, we can think of an object X as a "set" by taking "element" to mean "map S -> X" (where S is some other object). Allowing this object S to vary gives you precisely the presheaf represented by X, and Yoneda implies that this is enough information to determine X. This perspective allows for a very down-to-earth understanding of universal properties of limits, which I explain in my Kan Academy video 🙂
So, lets assume I manage to understand this information and transform it into knowledge. Could you suggest an example how it could be used to optimize code/program structure for example?
jPerhaps you could make a video relating this material to locale theory, perhaps with hints of how that leads to topos theory. My vague understanding is that a locale works on complete Heyting algebras (a frame), but the morphisms can't keep track of both top and bottom at the same time, So a geometric morphism forgets bottom, or something. I think a Heyting algebra is something like an unquotiented preset, while a Boolean algebra quotients by the equivalence relation of symmetric edges to get an antitisymmetric poset. This models extensionality, whereas Heyting algebras allow modeling intensional phenomena, like language.
12:30 Does this proof depend on LEM/DNE or is it possible to reformulate it to directly prove that x_1=x_2 rather than a proof by contradiction? In other words, is the proof fundamentally classical or can we make it constructive? I'm so used to category theory being constructive, so I was surprised to see a proof by contradiction here. *Edit:* 13:14 lmao!
I won't lie, I haven't thought very carefully through how everything changes without excluding the middle, though with a certain way of expressing "singleton" (as a set whose elements satisfy that any two of them are equal), I think the proof of uniqueness falls through without LEM.
@@SheafificationOfG yeah, that makes sense! it was so funny to see "huh, that proof seems to assume LEM" only to a few seconds later get the "You just assumed the Law of Excluded Middle! 🎵🎶H*ll naw, to the no no no 🎶🎵" jingle about something unrelated 😁
What a brilliant video! This got me thinking, why not define a set object as any object X that is isomorphic to the exp object [1, X]. Then we could define a coset as being any object isomorphic to the coexp object: Y ≈ ]Y, 0[
Interesting idea! The exponential object [W, X] is usually defined as a representing object for Hom((-) x W, X), so a co-exponential object ]A, B[ would be a corepresenting object for Hom(A, B + (-)). However, taking B to be the initial object, we see that ]A, 0[ always exists, and is isomorphic to A, just as [1, X] always exists and is isomorphic to X. Put differently, this just reiterates how "every object is set-like", and unfortunately using co-exponentials yields a somewhat unsatisfying conclusion that "every object is coset-like" as well.
I simply love your tag line "My university doesn't let me teach anymore, so the rest of the world is my victim now." Coz i too want to make the rest of the world my victim 😭 drowning them in my abstract games :D
why would preimages not compose well? given f(z) = y and g(y) = x, wouldn't the expected behavior of a composed preimage be the complete set of z that can produce an x, meaning the combined sets of preimages for every y in the preimage of the second operation?
That is definitely reasonable! Implicitly, you are extending f's preimage domain to the powerset so that it can be composed with the preimage of g. I just took a more uniform approach (by making this extension everywhere), which has the added bonus of having composition be just the usual composition of functions again (as intended).
First, we must consider it a set is even an operation to begin with. If it is, I'd say some sort of operator that goes from set, back to equation, or whatever the set's origin is.
I can usually follow without having to pause a lot, but damn this one I had to pause like every 2 seconds. Very interesting topic, but it felt more like you're reading a math paper out loud than a YT video, hopefully I can follow better next time! Thanks for making these tho
This is definitely a concept-heavy one, and doesn't admit a lot of helpful imagery (at least, not imo)! Maybe it would have been better as a blog post but blogs don't activate my creative neurons hahaha
Something that I've wondered about is, why is category theory so symmetrical in theory, but not in practice? We work a lot with Set, not CABA, cartesian closed categories, not cocartesian coclosed categories, toposes not cotoposes, categories where we might drop LEM but not ones where we might drop the principle of explosion. What does comathematics look like? a dual mathematics where the fundamental objects are cosets rather than sets? What does a perfectly symmetrical mathematics look like?
@@SheafificationOfG awesome, i look forward to it! Have you looked much into linear logic? That's where this line of questioning brought me but i didn't get very far with it
@@SheafificationOfG You'll probably want to look at the Chu construction. You get C and C^op embedding into a self dual Chu(C,d) with some choice of dualizing object d. Done with sets you get chu spaces which sort of have the feel of topological spaces. They're related to linear logic too, i think they can be a sort of categorical semantics? If you want a nice introduction to linear logic you can check out Mike Schulman's Affine Logic for Constructive Mathematics
@grudley Chu spaces aren't really the direction I want to take this theme (at least regarding asymmetry) but I already have them on my list of potential topics!
I mean I mean the very definition of set makes it likely that even some things considered opposites to it would be themselves set just transposed ones such as the set which excludes everything but I liked your point showing the issue with this in the middle of the video
I'll watch the video in the moment but my preliminary impulse is to say an exception. Its a single element, but partially belongs to more than one set without fully belonging to either.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Negation of the negation gives a positive -- Hegel. Truth is dual to falsity -- Propositional logic. Absolute truth is dual to relative truth -- Hume's fork. Truth is therefore dual. Union is dual to intersection -- Boolean algebra. Sine is dual to cosine or dual sine -- the word co means mutual and implies duality! Sinh is dual to cosh -- hyperbolic functions. Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry. Positive curvature is dual to negative curvature -- Gauss or Riemann geometry. There is a hidden dual basis in Riemann geometry. Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory. Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory. Subgroups are dual to subfields -- the Galois correspondence. The tetrahedron is self dual. The cube is dual to the octahedron. The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry. Objective information (syntax) is dual to subjective information (semantics) -- information is dual. Syntax is dual to semantics -- languages, communication, information. If mathematics is a language then it is dual. Large language models in neural networks are therefore using duality! Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex plane hence all numbers are dual. The integers are self dual as they are their own conjugates. Addition is dual to subtraction (additive inverses) -- abstract algebra. Multiplication is dual to division (multiplicative inverses) -- abstract algebra. Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics! "Always two there are" -- Yoda. Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
As someone with basically no knowledge of high level mathematics but some programming experience, I would imagine a set to be like an array system, and a coset to be like a tag system?
14:23 This seems wrong. Counterexample: Let I = N, S_i = {1}, S_i' = {i}, then the unions are {1} =/= N.
Crap, you're right; only finite unions and intersections are inherited from the power set. Completeness is another matter... thanks for pointing this out!
*Edit:* unfortunately, @yuvalpaz3752 showed also that A cannot be complete at all, meaning I have only provided an example of a (not complete) Boolean algebra with no atoms. For a *complete* example that's similar to this, you can take the algebra of Lebesgue measurable subsets of R, where two subsets are considered essentially the same if they differ by a set of Lebesgue measure zero. The argument follows the same general idea, but this example is actually complete as a Boolean algebra.
Sorry everyone!
@@SheafificationOfG I'm not sure exactly how you would define them, but I have a hunch that you'd be fighting the axiom of choice in favour of the axiom of determinacy to get everything you're looking for.
*Edit:* the content written below is nonsense, and is simply false (thanks @yuvalpaz3752 for calling me out!). Turns out the algebra A is not complete. :( Sorry everyone!
Ok, now that I have a bit of time, here's the construction (no need for AoC)! I'll describe joins, meets are analogous [dual, if you will].
For each element x of A, define its "ideal" [x] to be the set of all subsets of N that are essentially the same as some subset of x. You can identify x with its ideal (x is essentially the same as every maximal element of its ideal), and conversely, any subset of P(N) that is closed under inclusions and essential equality is the ideal of some (essentially unique) element of A).
Now, this may be a confusing set of words, but to compute the join of an infinite family (x_i)_i of elements x_i of A (indexed by some infinite set I), do the following.
1. For every finite subset J of I, compute the union u_J of the x_j for j in J (this can be done set-theoretically).
2. For each u_J, compute its ideal [u_J].
3. Since each ideal [u_J] is a subset of P(N), take the union of these [u_J] over all finite subsets J of I. This is a new subset U of P(N).
4. U is closed under inclusions and essential equality, and is therefore the ideal of some (essentially unique) element of A. This element is precisely the join of the x_i.
[In fancy categorical nonsense, the above strategy is how to compute filtered colimits in A viewed as a category, and since we can compute finite coproducts (unions), arbitrary unions can be written as a filtered colimit of finite unions.]
@@SheafificationOfG your definition is problematic, in step 4 you took union of "ideal"s and claim this to be an "ideal".
Look at x_i={p_i^n | n in N} for p_i the i-th prime. Take the union of [u_J] for u_J as you defined, call it U. If U=[A] then A contains all but finitely many elements from x_i for each i, let z_i the first power of p_i that is in A and define B to be {x in A | x is not equal to z_i for any i}.
A simple argument shows that each u_J is in [B], in particular U is a subset of [B], but A is not in [B] by definition.
Edit: a similar argument shows that P(N)/fin is not complete under any sensible order
@yuvalpaz3752 crap you're absolutely right. P(N)/fin is not complete at all... my bad.
(Though in defense of the term "ideal", these are ideals in the order-theoretic sense... my mistake was that U is not a principal ideal, as shown by your argument.)
A mathematician is someone who turns coffee into theorems.
A comathematican is someone who turns cotheorems into ffee.
And an engineer turns theorems into coffee. Yet more duality?
So a journal editor?
@@undefinednan7096Converts theorem to toilet paper. So a co-editor Toilet papers a theorem to Nverts.
coffee is for losers
OK now do smologists
"Hello id like THE OPPOSITE OF A SET" sentences DREAMED UP by the UTTERLY DERANGED
A video about sets, no chance this is more complicated than a math 101 class.
Edit: I lasted 1 minute and 10 seconds.
I lasted only 1 minute :( also I watched the entire video
Video was too good, only lasted 18 seconds
the new try not to cum math challenge is crazy
6 minutes :(
You're a champ I just got my degree a few months ago and only lasted 47 seconds 😭😭
I'm so glad I coauthored this video.
Are you going to corelesse covideo?
@@AntoshaPushkin you're reading it
No, the covideo coauthored coyou.
Thank you for your videos and vering untless ol ncepts mprehensibly!
Umm?
@@gregoryfenn1462 use the complementary words
He used the anti-cofunction of each word lol
mpregensively
love love LOVE this channel. So glad I found this while I'm still in the knows-enough-about-category-theory-to-follow-the-video-but-not-enough-to-already-know-everything-in-the-video stage. Awesome.
The title made me think that the video is more digestible. I lasted 4 minutes.
1 min 10 seconds for me 💀
@@alexandergu7797 You guys missed the best memes.
I am constantly amazed by how well you blend category theory and internet memery; I keep dipping my toes into category theory and suddenly finding myself over my head, but you're one of the best life rafts (to really drag out this analogy) that I've found. Thank you!
Minor nitpickery: at 2:53 I think you have a couple of the composition formulae wrong. sr needs to be s, not r (both are arrows from top-middle to top-right) and vs and v are likewise between different objects of C; I'm pretty sure that has to be vs=u almost definitionally since with {0} a terminal object every arrow from top-middle to bottom is just u.
Thanks so much! (And yeah, you're right! Sloppy me, I think I changed notation in the diagram and forgot to update the equations...)
I didn't co-liked the video, I just co-understood it.
It understood you?
@@drdca8263 "Co-it" and "co-you" actually
@@andrasfogarasi5014 Thank you for the correction
@@drdca8263 omg both you and op made me laugh so hard xD
Not assuming the Law of Excluded Middle allows me to neither not like it nor not not like this video without any ambiguity. Sort of.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
This takes me back to mathematical logic in my first year of uni. It's really hard to grasp how, even for very philosophically abstract considerations you can run into fun math problems (like the combinatorial argument you showed at the beginning for the difficulty of finding which functions can be preimages).
1:01 the set theorists are the ones who will hunt you down. in mode theory you pretend there is a university (usually called a monster) which is a set all of the time.
That's a good point. Guess *I'm* the one who's gonna get in trouble with the model theorists...
In model theory, a set in one model of set theory may not be a set in another. And also, the elements of a set aren't property of the set itself but of the set membership relation. So in model theory, there's no such thing as small or large collections, only elements and non-elements of a model. Actually there is if one refers to the meta-language (e.g. elementary class), but that's more like the concern of set theorists.
@@Noname-67 no, i didn’t mean the model theory of set theory. i meant, you fix a cardinal κ that’s large enough for your whatever you are trying to do, you fix a “monster” model M of your theory that is κ-saturated and you pretend M is the universe.
you wouldn’t do this with ZFC because it has no good model theoretic properties (you can not even find good ways of completing it), but you would with other theories (like algebraically closed fields or something).
i'm very early on in this but i have some conflicted feeling, namely about considering the defining property of a set as being an object in the category of sets *and functions as morphisms*.
maybe it's too pedantic but from the perspective of sets and sets alone, it's not apparent that functions are singled out as special operations... relations seem just as fundamental. and Rel is self-dual!
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
I didn't co-understand this, but found it co-mpletely co-mpelling and co-nfusing in the most co-medic way. Thank you.
So glad I found your channel Sheafification of G! This was something I actually thought about briefly but never bothered to try; seeing the answer put together in such an entertaining way was a real joy to watch! You got my sub!
I think I can categorically state that yours is the most hilarious abstraction astronautics on UA-cam. (Hmm, is astronautics cospelunking?)
UA-cam forces me to write a comment on the donation apparently, but guess I can say that I really appreciate your videos man💯
Hahaha thanks fam!!
i have literallt no idea whats going on. wtf is happening???? im at 2:04
You mean C^{op} or the page joke?
just vibe and enjoy your youtube algorithm being irrevocably degraded with mental illness posing as "math"
imma pretend i understood something
Same
it's like a brain massage
Opposite of a "set"? Well of course it's a "dynamic"!
Still waiting for a video on Type Theory…
I probably could understand this, if I turned on my brain. Instead I'll just enjoy some fancy animations and go "yup, makes sense" at the end. It's a nice video, and somehow gives me the vibes of a tech DIY youtuber showing off a new toy they made, except it's a mathematician creating new branches of math on a napkin. I have a hunch that a lot of effort went into discovering every line of equations that was shown. I wish I had a solid enough grasp of the field to do the same. Seems like magic. Rock on, science man :)
Well, in reality, this result is well-known (among category theorists), so it didn't take much ingenuity on my part 😉
I've never tried to understand boolean algebras as I thought they were not very fun. That is until I see this video. I love the way you presentet this, especially that the term boolean algebra only appeared halfway into the video so I didn't have time to run away lol
Thanks so much!
Hell yeah, the graph + directional arrow representation made this easy to grasp. Also it brought up Feynman diagrams. Nice.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
6:11 - If g is from Y to X, then the preimage function should be from P(X) to P(Y).
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
I've got into Awodey a few weeks back and this was just that one other angle of view that made me understand a bunch of things. Thank you
I am a simple man with a Master Degree in category theory.
I see the notation for sheafification of Sheaves in Geometry and Logic, I press subscribe.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Statisticians in SHAMBLES after this one
Wtf this vid made me rethink my mathematical capabilities 💀
This was great fun! .... I'm *so* glad I got into Physics!
"You might have noticed that, in some bold and brash way, every power set contains its original set: the elements of the original set each define a unique one-element subset."
I got flashbacks to learning about NFU, where a power set can be smaller than the original set because the injection x ↦ {x} can't be created
i understood what a lot of these words mean!!! but i counderstood the meaning of them put together
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
As an undergrad maths student, just wanted to say I absolutely love your videos - perfect pacing, topics, humour, random shitty osu! references - honestly, perfect
Every morning I throw youtube on while I eat my breakfast. Today I had a bowl of still not understanding category theory lmao
Does the co-CABA of all co-CABAs co-ntain itself?
The quick reference to Papers Please was awesome
And here I was thinking a coset is the set of all objects not contained in the original set, such that set union coset = universe.
I am a new fan of this channel.
I’m co-proud of my ability to understand this admittedly co-boring video
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
gen z version of 3blue1brown
-3yellow-1navy
It's after midnight on a Friday... I guess Saturday now, and I'm watching this after drinking whiskey for the past few hours. Not entirely sure why this one came up, but it made sense and is well done. Thanks for making this and now it is time to sleep!
but what's the co-opposite of a set?
Opposite is a (contravariant) functor, and the opposite of a functor is the same functor but on the opposite categories. Since Opposite is a functor mapping a category to its opposite, co-Opposite is a functor mapping the opposite a category to said category. This is simply the same as Opposite. In other words, the opposite of Opposite is simply the Opposite. Which means that the co-opposite of a set is simply "coset".
At first I was confused by this, since I’d assume the co-opposite of set would be set itself. Here’s the analogy that worked for me if anyone else is confused.
Think of involutions (functions that are their own inverse), like “negative”. The function “negate” sends positive numbers to negative ones and vice-versa (with 0 staying where it is). The opposite of this sends negative numbers to positive ones and vice-versa… which is just the same as negating. This is why negative-negative makes positive.
Notably, finding the inverse of a function reverses _the behaviour of the function_ but _doesn’t affect its input._
Similarly, applying “co” to a functor reverses its behaviour, not its input. So if a functor is its own cofunctor, its behaviour is unchanged. The “opposite” function can be thought of like “negate”, since opposite-opposite is just no change.
Therefore co-opposite is just opposite, and applying it to set gives back the opposite of set, analogously to how -^(-1) x = -x and not +x. Neat!
Incredibly valuable. ST is fascinating
This is very cool. A simpler (though probably less interesting) answer is that the opposite of Set is the category whose objects are sets and whose morphisms are injective and surjective relations.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
13:08 I don't get why (Assuming excluded middle) the video doesn't talk about complements as it is a part of the definition of boolean algebras. for inverse images g, g(A\B) = g(A)\g(B) and X\(A U B) = X\A intersect X\B.
Edit: I think I get it, kinda got swept under the rug but since we are only considering the complete lattice of power sets under set inclusion, well that particular example is orthocomplemented. Though not all complete lattices are orthocomplemented. What is kinda interesting here is that homomorphisms of completed lattice can preserve orthocomplmention somehow. probably something like the homomorphism preservation theorem?
Ayo there is a spooky amount of overlap between my coordinate function-space, and this coset stuff you made. 😳
I'm currently working on it with professors and it will likely be published to academia in the coming weeks.
I'll be sure to shout you out, Given you managed to scrape up just some of this coordinate function-spaces properties on your own.
I'll also Email send you the paper when i publish it to keep you up to date. 😁👍
Looking forward to it!
needs to start adding prereqs to his videos
You can also get some notion of coset from linear type theory. There you can get back to usual type theory via !. In some contexts ! Is the cofree (cocommutative?) comonoid functor, which implies that regulat types/sets are those linear types which are (cocommutative) comonoids. Dually cosets would be (commutative) monoids, but over ⅋ (par) not the usual product. But because linear logic is self dual they just correspond to the negaton of regular sets, which is a bit boring.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
2:27 you say “some people, like analysts…” I got my PhD in mathematical analysis, I can assure you we call ourselves “Analysists” not “Analysts” as the latter is a different job entirely.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
This is spectacular, especially where you characterize the maps that are preimages
*Ventilateur * Merci d'avoir lu ce commentaire
This might be the most intriguing title ive ever seen
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
I'm dying from laughter, thanks for making these! LMAO
category theorists should start with this, it's the first thing that has actually gotten me to see the point of category theory
this is mindblowing
I have never felt a more visceral biological impulse to point threateningly at someone and yell "NEEEEERD!"
Learning that I've spent too much time studying mathematics because I can guess what the answer is going to be after a couple of minutes is a fun way to start my morning
this was such a cool video about something i've been thinking about myself! specifically i was thinking about an Ind-dual to condensed sets. if you Ind-complete (rather than Pro-complete) the category of finite sets you get countable sets, and product preserving presheaves on these form the so called "bornological topos". i feel like an algebraic method could be used on these similar to condensed sets. this video is inspiring me to give this another go
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
10:18 Ah, my old friend and nemesis: THE AXIOM OF CHOICE!!!!
I must admit, this one went over my head
This was ❤. Now I see stone duality more clearly too.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
my favorite non-example from software is that in linux you kexec is not called coreboot, that's something else entirely
Are copilot on planes to back them from the gate?
Very nice vid thank you :D I'm new to category theory and while I understand the yoneda lemma superfically I don't have good intuition for it. Could someone elaborate on the statement G makes at 1:52, about how all objects are set objects? I don't see how this follows from the yoneda lemma but it seems like it could be a helpful fact to think about general categories.
Glad you liked the vid!
Essentially, this is an interpretation of how the Yoneda embedding is fully faithful, or that objects are uniquely determined up to isomorphism by their representable presheaf.
Basically, we can think of an object X as a "set" by taking "element" to mean "map S -> X" (where S is some other object). Allowing this object S to vary gives you precisely the presheaf represented by X, and Yoneda implies that this is enough information to determine X.
This perspective allows for a very down-to-earth understanding of universal properties of limits, which I explain in my Kan Academy video 🙂
So, lets assume I manage to understand this information and transform it into knowledge. Could you suggest an example how it could be used to optimize code/program structure for example?
Subscribed!
Nice! I didn't know about this.
Is there a connection, in the end, w/ the usage of the word coset in group theory?
I understand just enough of this video that its super interesting lmao. Thank you for making it :))
jPerhaps you could make a video relating this material to locale theory, perhaps with hints of how that leads to topos theory. My vague understanding is that a locale works on complete Heyting algebras (a frame), but the morphisms can't keep track of both top and bottom at the same time, So a geometric morphism forgets bottom, or something. I think a Heyting algebra is something like an unquotiented preset, while a Boolean algebra quotients by the equivalence relation of symmetric edges to get an antitisymmetric poset. This models extensionality, whereas Heyting algebras allow modeling intensional phenomena, like language.
12:30 Does this proof depend on LEM/DNE or is it possible to reformulate it to directly prove that x_1=x_2 rather than a proof by contradiction? In other words, is the proof fundamentally classical or can we make it constructive? I'm so used to category theory being constructive, so I was surprised to see a proof by contradiction here.
*Edit:* 13:14 lmao!
I won't lie, I haven't thought very carefully through how everything changes without excluding the middle, though with a certain way of expressing "singleton" (as a set whose elements satisfy that any two of them are equal), I think the proof of uniqueness falls through without LEM.
@@SheafificationOfG yeah, that makes sense!
it was so funny to see "huh, that proof seems to assume LEM" only to a few seconds later get the "You just assumed the Law of Excluded Middle! 🎵🎶H*ll naw, to the no no no 🎶🎵" jingle about something unrelated 😁
what is a dual construction for subobject classifier in set^op?
Omg this is so awesome!!
If a monad is a burrito then is a comonad an enchilada? lol
What a brilliant video! This got me thinking, why not define a set object as any object X that is isomorphic to the exp object [1, X]. Then we could define a coset as being any object isomorphic to the coexp object: Y ≈ ]Y, 0[
Interesting idea! The exponential object [W, X] is usually defined as a representing object for Hom((-) x W, X), so a co-exponential object ]A, B[ would be a corepresenting object for Hom(A, B + (-)). However, taking B to be the initial object, we see that ]A, 0[ always exists, and is isomorphic to A, just as [1, X] always exists and is isomorphic to X.
Put differently, this just reiterates how "every object is set-like", and unfortunately using co-exponentials yields a somewhat unsatisfying conclusion that "every object is coset-like" as well.
Now I am starting to think a coconut might just be a nut.
Do co-algebras next, if you haven't already! 😉
Some great memeification in this one! 👍
I simply love your tag line "My university doesn't let me teach anymore, so the rest of the world is my victim now."
Coz i too want to make the rest of the world my victim 😭 drowning them in my abstract games :D
why would preimages not compose well? given f(z) = y and g(y) = x, wouldn't the expected behavior of a composed preimage be the complete set of z that can produce an x, meaning the combined sets of preimages for every y in the preimage of the second operation?
That is definitely reasonable!
Implicitly, you are extending f's preimage domain to the powerset so that it can be composed with the preimage of g. I just took a more uniform approach (by making this extension everywhere), which has the added bonus of having composition be just the usual composition of functions again (as intended).
There was a lot that I didn't follow but thanks for the humour.
First, we must consider it a set is even an operation to begin with.
If it is, I'd say some sort of operator that goes from set, back to equation, or whatever the set's origin is.
I can usually follow without having to pause a lot, but damn this one I had to pause like every 2 seconds. Very interesting topic, but it felt more like you're reading a math paper out loud than a YT video, hopefully I can follow better next time! Thanks for making these tho
This is definitely a concept-heavy one, and doesn't admit a lot of helpful imagery (at least, not imo)! Maybe it would have been better as a blog post but blogs don't activate my creative neurons hahaha
The irony of a category theorist classifying the universe of sets
every time I learn CT I have the voice of Bruno Powroznik in my head
intuitionistic set theory! what even is a non-empty / infinite set ??
Something that I've wondered about is, why is category theory so symmetrical in theory, but not in practice? We work a lot with Set, not CABA, cartesian closed categories, not cocartesian coclosed categories, toposes not cotoposes, categories where we might drop LEM but not ones where we might drop the principle of explosion. What does comathematics look like? a dual mathematics where the fundamental objects are cosets rather than sets? What does a perfectly symmetrical mathematics look like?
This is a topic I actually hope to touch on more in the future!
@@SheafificationOfG awesome, i look forward to it! Have you looked much into linear logic? That's where this line of questioning brought me but i didn't get very far with it
@@grudley Nah, I'm honestly not much of a logician. When I get more serious about duality we'll see where the wind takes me haha.
@@SheafificationOfG You'll probably want to look at the Chu construction. You get C and C^op embedding into a self dual Chu(C,d) with some choice of dualizing object d. Done with sets you get chu spaces which sort of have the feel of topological spaces. They're related to linear logic too, i think they can be a sort of categorical semantics? If you want a nice introduction to linear logic you can check out Mike Schulman's Affine Logic for Constructive Mathematics
@grudley Chu spaces aren't really the direction I want to take this theme (at least regarding asymmetry) but I already have them on my list of potential topics!
Ngl, this one flew right over my head.
I mean I mean the very definition of set makes it likely that even some things considered opposites to it would be themselves set just transposed ones such as the set which excludes everything but I liked your point showing the issue with this in the middle of the video
I'll watch the video in the moment but my preliminary impulse is to say an exception.
Its a single element, but partially belongs to more than one set without fully belonging to either.
The opposite of a set is a partition: one unites elements, the other divides them.
I hear the question "what is the opposite of a set" and my mind asploded
I’m happy people on UA-cam actually care enough about this to make a video🥹 Hope people watch and care about learning more math.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
this was a great video to pretend i understood while watching
Wait so if we have a coalgebra and a covector then do we have a cocalculus? Would that just be discrete math? Just a random thought lol
I saw opposite and set and I was like here we go for more category theory
i cant waot for like 4 years from now when i finally understand everything in this video
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
13:15 doctor is it serious? it keeps happening to me
By opposite, do you mean negation?
Negation of the negation gives a positive -- Hegel.
Truth is dual to falsity -- Propositional logic.
Absolute truth is dual to relative truth -- Hume's fork.
Truth is therefore dual.
Union is dual to intersection -- Boolean algebra.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality!
Sinh is dual to cosh -- hyperbolic functions.
Contravariant is dual to covariant -- vectors or functors or a dual basis in Riemann geometry.
Positive curvature is dual to negative curvature -- Gauss or Riemann geometry.
There is a hidden dual basis in Riemann geometry.
Categories (syntax, form) are dual to sets (semantics, substance) -- Category theory.
Sets are dual to cosets or dual sets, domains are dual to co-domains -- Group theory.
Subgroups are dual to subfields -- the Galois correspondence.
The tetrahedron is self dual.
The cube is dual to the octahedron.
The dodecahedron is dual to the icosahedron -- the Platonic solids in geometry.
Objective information (syntax) is dual to subjective information (semantics) -- information is dual.
Syntax is dual to semantics -- languages, communication, information.
If mathematics is a language then it is dual.
Large language models in neural networks are therefore using duality!
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (syntropy) is dual to differentiation (entropy) -- abstract algebra or the 4th law of thermodynamics!
"Always two there are" -- Yoda.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
As someone with basically no knowledge of high level mathematics but some programming experience, I would imagine a set to be like an array system, and a coset to be like a tag system?