Category Theory III 3.2, Monad Algebras

Algebras that are compatible with a monad

[skibaa]

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

[siyuanchen]

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.

[twoscoops]

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.

[JoelSjogren]

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