External Explanations

Contravariant

• George Wilson's Contravariant Functors: The Other Side of the Coin

Comonad

• Chris Penner's Comonads by Example (I highly recommend it).
• Gabriel Gonzalez's You Could have Invented Comonad
• Phil Freeman's Comonads as Spaces
• Nihil Shah's Cofree Comonads and their Uses

IndexedMonad (i.e. IxMonad)

An IndexedMonad uses the compiler to ensure:

1. monadic computations occur in the proper order
2. one monadic computation will change what the final output will be

Links:

• Kwan's Indexed Monad overview
• What I wish I knew when Learning Haskell's Indexed Monad section
• Justin Woo's Using IxMonad to enforce good hamburger building in PureScript

Real-world problem where it would help: purescript-selda, a library that allows one to write type-safe database queries, would prevent some aggregate queries from working when they are invalid. The (thesis' (PDF)) relevant sections are 2.3, 2.4, and 5 (paragraph 5 and down).