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I've seen the term Free Monad pop up every now and then for some time, but everyone just seems to use/discuss them without giving an explanation of what they are. So: what are free monads? (I'd say I'm familiar with monads and the Haskell basics, but have only a very rough knowledge of category theory.)
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A monad is an algebraic structure in category theory, and in Haskell it is used to describe computations as sequences of steps, and to handle side effects such as state and IO. Monads are abstract, and they have many useful concrete instances. Monads provide a way to structure a program.
In terms of OO programming, a monad is an interface (or more likely a mixin), parameterized by a type, with two methods, return and bind that describe: How to inject a value to get a monadic value of that injected value type; How to use a function that makes a monadic value from a non-monadic one, on a monadic value.
A functor is an interface with one method i.e a mapping of the category to category. Monads basically is a mechanism for sequencing computations. A monad is a way to wrap stuff, then operate on the wrapped stuff without unwrapping it.
The state monad is a built in monad in Haskell that allows for chaining of a state variable (which may be arbitrarily complex) through a series of function calls, to simulate stateful code.
Here's an even simpler answer: A Monad is something that "computes" when monadic context is collapsed by join :: m (m a) -> m a (recalling that >>= can be defined as x >>= y = join (fmap y x)). This is how Monads carry context through a sequential chain of computations: because at each point in the series, the context from the previous call is collapsed with the next.
A free monad satisfies all the Monad laws, but does not do any collapsing (i.e., computation). It just builds up a nested series of contexts. The user who creates such a free monadic value is responsible for doing something with those nested contexts, so that the meaning of such a composition can be deferred until after the monadic value has been created.
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