Category Theory For Beginners: Duality And Functors

[@englemanart]

Great series. Going through it a second time, thanks!

[@RichardSouthwell]

The next video `Category Theory For Beginners: Universal Properties' is scheduled to become public on June 8. Then `Category Theory For Beginners: Natural Transformations' will become public on June 12.

[@JaGWiREE]

Very nice series. Thanks for the periodic updates, I am in a reading group starting Riehl's book on the 6th and this series has made for great motivation and exposure.

[@gucker]

Thanks for all the work you devote to these lectures. They are very interesting and educating!

[@RichardSouthwell]

@@JaGWiREE I'm really glad they are helpful

[@JaGWiREE]

@@RichardSouthwell your other series on topology has been incredibly useful for me as well. Thanks so much for this well thought out pedagogical material

[@monktastic1]

Lovely series! You are a fantastic teacher! Just as a small correction (around 7:30), the empty set isn't contained in all other sets but is a subset of them.

[@the_allucinator]

By illustrating well how a disjoint/discriminated union works, it gave me a eureka moment. Thank you! For example, imagine a 3x3 grid. I want to index each row and columns from 1 to 3 like this: (x, y) where x is the base of the grid which is 3 and y is the nth row or nth column. (x,y) can be read as row/column y of x-square. Now if we apply to a 3-square grid. The set of row indices of 3-square would be {(3,1), (3,2), (3,3)} and the set of column indices would be {(3,1), (3,2), (3,3)} also. The problem is that the set of row indices and column indices are the same. Now your video gave me an idea about disjoint unions. If I were to distinguish between rows and columns, I can just use disjoint unions. I wrote a book about representing a 2D array in 1D array using square grids in a semi-formal math tone. (I am not a mathematician, but I am a software developer. I have little to average math knowledge/proficiency.) Here's a Github repo that holds my book: github.com/makoru-hikage/sasquare-docs I will revise my book soon.

[@user-tk2jy8xr8b]

So, do you just omit (A, Bbar) and (Abar, B) from CxC category in explanations for the sake of diagrams simplicity?

[@MegaPegazuz]

Those videos are amazing ! Thank you very much !

[@gucker]

Thank you for the another brilliant lecture!

[@youngmong6966]

It seems to me that the functor introduced here is something like a isomorphism. Is there something like isomorphism and homomorphism in different functors?

[@Quux_Music]

You've got a mistake at 1:06:18, you've written F((id_A, id_Abar)) = F(id_AxAbar), where it should be F((id_A, id_Abar)) = id_F(A, Abar) = id_AxAbar. Great video though!!

[@NaderHGhanbari]

Doesn't the category have to be Cartesian closed for a product functor to exist from CxC -> C ?

[@anshanshtiwari8898]

8:50 ex falso quodlibet is the most weird thing to wrap one's head around. I just convince myself that this is just an assumption that renders everything else consistent and makes life easier.

[@hyperduality2838]

Functions have goals, targets, objectives or intentions, teleology, morphisms. Teleological physics is dual to non-teleological physics. Randomness (entropy) is dual to order (syntropy) Projections = predictions, anticipations, expectations, thesis, hypothesis or prophecy. The future is dual to the past. Observers predict or project the future and remember the past, time duality, time asymmetry. Time is dual, space is dual and time is dual to space -- Einstein

[@hyperduality2838]

Energy is dual to mass -- Einstein Dark energy is dual to dark matter Syntropy is dual to increasing entropy -- The 4th law of thermodynamics Syntropy is the integration, convergence or reunion of information to form predictions, expectations or priors. Entropy is the differentiation, divergence or splitting of information into new states. Integration is dual to differentiation, divergence is dual to convergence Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought Potential energy is dual to kinetic energy, energy is inherently dual. Duality is energy, energy is duality and duality is being conserved -- The 5th law of thermodynamics.

[@hyperduality2838]

"Perpendicularity in hyperbolic geometry is measured in terms of duality"

[@hyperduality2838]

The subjective is dual to the objective, the relative is dual to the absolute, dependence is dual to independence. Complexity is subjective it is dependent upon the observer's perspective, simplicity is objective it is independent of the observer's perspective. Complexity is dual to simplicity --> Syntropy is dual to increasing entropy. Certainty is dual to uncertainty, the Heisenberg certainty/uncertainty principle The observer is dual to the observed. Points, singularities or objects in Category theory are inherently dual they can only be defined with reference to or in terms of other points, singularities, objects. Generalization is dual to localization. Duality creates reality.

[@xyzct]

I laud you. (See what I did there?)

[@hyperduality2838]

@@xyzct No! Homology is dual to co-homology. Elliptic curves are dual to modular forms -- Fermat's last theorem. Its physics and the philosophy supports the physics! Same is dual to different. You are the same person now but different compared to a few hours ago. The passage of time means that you are a different person but still the same. Personal identity or the self is actually dual -- time duality, the future is dual to the past. Duality is important because the conservation of duality (energy) will be known as the 5th law of thermodynamics, energy is duality, duality is energy. Potential energy is dual to kinetic energy. Electro is dual to magnetic, light or photons are dual. "The art of progress is to preserve order amid change and to preserve change amid order" -- the duality of Alfred North Whitehead, philosopher. Being is dual to non-being creates becoming -- Plato. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Hegel's cat:- Alive (thesis, being) is dual to not alive (anti-thesis, non being) -- Schrodinger's or Plato's cat. Schrodinger's cat is based upon the metaphysics of Hegel, Kant, Descartes, Plato, Aristotle etc. First or efficient cause is dual to final cause -- Aristotle, material and formal cause are by-products of conserving this duality, a Janus hole. Concepts are dual to percepts -- the mind duality of Immanuel Kant.

[@hyperduality2838]

@@xyzct Points (objects) are dual to lines (representations) -- the principle of duality in geometry. Categories and concepts have their foundation in Immanuel Kant's "The Critique of Pure Reason".

[@xyzct]

@@hyperduality2838, your comments are excellent. And what do you say of "weak duality" -- such as between the lattice of subsets and the lattice of quotient sets?

[@hyperduality2838]

@@xyzct You have got me there. Could you please explain this to me! My background is in physics not mathematics. When it comes to sets I know the following:- Union is dual to intersection. If you have a good video on youtube about this I can give it a watch and see if I can understand it.