Reader monad, categorically - Haskell

How would I describe monads such as MonadReader or MonadState or even Applicative in the language of category theory? When searching for these I only find their functional programming description (even Applicative, which I would expect to be easy to find a categorical description).

applicative is a "monoidal functor" afaiu

Cool! For future reference:

• http://www.staff.city.ac.uk/~ross/papers/Applicative.pdf
• https://bartoszmilewski.com/2017/02/06/applicative-functors/

Still, it would be interesting to figure if there is a good description of MonadReader in the language of category theory.

something like a monad in the Hom category

I guess Reader is just a function, and MonadX constraints are sort of "proofs" of monad transformers being composable with X monad?

right so the question is whether @Pedro Minicz meant ReaderT or specifically MonadReader

MonadReader specifically.

Torsten Schmits said:

something like a monad in the Hom category

Which hom-category? As far as I know, a hom-category is the category between 0-cells in a 2-category.

I believe I haven't heard anyone talk about about the "2-category of endofunctors," which is the context in which monad are usually discussed. What should the 2-cells be?

Check out the last issue of The Monad Reader, published in March 2009, which answers the question "where do the common fundamental monads of Haskell, such as State, Reader, Writer, List, and Maybe, come from?" in the article by Derek Elkins.

https://wiki.haskell.org/wikiupload/8/85/TMR-Issue13.pdf