What is indexed monad?

Question

What is indexed monad and the motivation for this monad?

I have read that it helps to keep track of the side effects. But the type signature and documentation doesn\

Answer 1

It may be important to take a look how indexing is used in dependent types (eg in agda). This can explain how indexing helps in general, then translate this experience to monads.

Indexing permits to establish relationships between particular instances of types. Then you can reason about some values to establish whether that relationship holds.

For example (in agda) you can specify that some natural numbers are related with _<_, and the type tells which numbers they are. Then you can require that some function is given a witness that m < n, because only then the function works correctly - and without providing such witness the program will not compile.

As another example, given enough perseverance and compiler support for your chosen language, you could encode that the function assumes that a certain list is sorted.

Indexed monads permit to encode some of what dependent type systems do, to manage side effects more precisely.