Category Theory III 3.2, Monad Algebras

[@skibaa1]

19:03 when you have object (a, sigma) and f: a->b how exactly you construct gamma:Fb->b ? Lifting f to Fa->Fb does not help you at all, if you need to start at Fb and to construct b

[@skibaa1]

probably the morphisms in the category of algebras should be also pairs of functions, one for converting the carriers and another for converting the evaluator

[@skibaa1]

also in general case in an algebra category there may be different objects (b, gamma) and (b, delta) with the same carrier but different evaluators

[@DrBartosz]

You start with two arbitrary algebras and ask the question: Are there any morphisms between them? An algebra morphism is defined as a morphism between the carriers that makes the diagram commute.

[@twoscoops9132]

Awesome stuff! I hope next time you'll shed some light on that wonderful category of *monadic adjunctions*. How exactly can the Eilenberg-Moore adjunction be seen to be the initial object, and how can the Kleisli be seen to be the final? And what other light can be shone on them and how they relate; in particular, how can the Kleisli be seen to be the full subcategory of Eilenberg-Moore *free algebras*? (And maybe if we're real good, give us some insight on the dual constructions!) This is something that's always been hard for me to find explained adequately, and I'm dying to see! It would open up a lot of very important and very fun stuff.

[@ShimshonDI]

What exactly are the objects in a category of adjunctions, and what are the morphisms? I know you defined an adjunction in video II: 5.2 as an adjoint pair of functors, together with a unit and counit (which are natural transformations), such that the triangle identities hold. I guess I'm not comfortable yet viewing an adjunction (i.e. all those things together) as a single object, and reasoning out what the morphisms would be between them.

[@DrBartosz]

See the Wikipedia article: en.wikipedia.org/wiki/Monad_(category_theory)#Monads_and_adjunctions

[@ShimshonDI]

Thank you. Since I'm finding category theory concepts so difficult at this level (but finding it so interesting), I'm trying to get a sense of what would be best: building more programming and/or other math background first, or maybe getting the right textbook and actually working exercises. What would you recommend?

[@DrBartosz]

Everybody is different, so there's no universal answer to it. I personally like to learn by teaching, either blogging or lecturing. I learn most when I have to explain things to others.

[@ShimshonDI]

I like teaching too. btw at least some of my remaining confusion has been cleared up in video III: 4.2, where you introduce Wikipedia's notation of a superscript or subscript T. A discrepancy: you say the initial object in the category of adjunctions generating a particular monad T corresponds to the Eilenberg-Moore category, whereas Wikipedia shows it corresponding to the Kleisli category.

[@shouya]

When Dr. Milewski talks about the composition of two adjunctions (C,D;L,R) and (D,E;L',R'), composing these adjunctions gives a single adjunction (C,E;L'∘L,R∘R'). I hope my unexplained notation here is clear to you. In fact it might easier to see it yourself by drawing the diagram on a paper.

[@siyuanchen659]

10:00 you said that "Eilenberg-Moore category and the adjunction related to it is the inital object in this category of adjunctions", but in your book p.381, you mentioned "Eilenberg-Moore adjunction is the terminal object." After do some google search, I am sure the book is correct.

[@DrBartosz]

Yes, you're right about that.

[@JoelSjogren0]

I'm going to save this treat for tomorrow :P :D

[@kinbolluck476]

👻