Monads and other dark magic for Elixir
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Witchcraft is a library providing common algebraic and categorical abstractions to Elixir. Monoids, functors, monads, arrows, categories, and other dark magic right at your fingertips.

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Table of Contents

Quick Start

def deps do
  [{:witchcraft, "~> 1.0"}]

# ...

use Witchcraft

Library Family

Quark    TypeClass
    ↘    ↙
Name Description
Quark Standard combinators (id, compose, &c)
TypeClass Used internally to generate type classes
Algae Algebraic data types that implement Witchcraft type classes


Beginner Friendliness

You shouldn't have to learn another language just to understand powerful abstractions! By enabling people to use a language that they already know, and is already in the same ballpark in terms of values (emphasis on immutability, &c), we can teach and learn faster.

As much as possible, keep things friendly and well explained. Concrete examples are available via doctests.

Consistency & Ethos

Elixir does a lot of things differently from certain other functional languages. The idea of a data "subject" being piped though functions is conceptually different from pure composition of functions that are later applied. Witchcraft honours the Elixir/Elm/OCaml way, and operators point in the direction that data travels.

Some functions in the Elixir standard library have been expanded to work with more types while keeping the basic idea the same. For example, <> has been expanded to work on any monoid (such as integers, lists, bitstrings, and so on).

All operators have named equivalents, and auto-currying variants of higher order functions are left at separate names so you can performance tune as needed (currying is helpful for more abstract code). With a few exceptions (we're looking at you, Applicative), pipe-ordering is maintained.


Convincing a company to use a language like Haskell or PureScript can be challenging. Elixir is gaining a huge amount of interest. Many people have been able to introduce these concepts into companies using Scala, so we should be able to do the same here.

All functions are compatible with regular Elixir code, and no types are enforced aside from what is used in protocol dispatch. Any struct can be made into a Witchcraft class instance (given that it conforms to the properties).

Type Class Hierarchy

Semigroupoid  Semigroup  Setoid   Foldable   Functor -----------┐
     ↓           ↓         ↓         ↓      ↙   ↓   ↘           |
  Category     Monoid     Ord    Traversable  Apply  Bifunctor  |
     ↓                                       ↙    ↘             ↓
   Arrow                            Applicative   Chain       Extend
                                             ↘    ↙             ↓
                                              Monad           Comonad

Having a clean slate, we have been able to use a clean of typeclasses. This is largely taken from the Fantasy Land Specification and Edward Kmett's semigroupoids package.

As usual, all Applicatives are Functors, and all Monads are Applicatives. This grants us the ability to reuse functions in their child classes. For example, of can be used for both pure and return, lift/* can handle both liftA* and liftM*, and so on.

Import Chains

It is very common to want to import a class and all of its dependencies. You can do this with use. For example, you can import the entire library with:

use Witchcraft

Or import a module plus all others that it depends on:

use Witchcraft.Applicative

Any options that you pass to use will be propagated all the way down the chain:

use Witchcraft.Applicative, except: [~>: 2]

Some modules override Kernel operators and functions. While this is generally safe, if you would like to skip all overrides, pass override_kernel: false as an option:

use Witchcraft.Applicative, override_kernel: false

# Or even

use Witchcraft, override_kernel: false

Writing Class Instances

How to make your custom struct compatible with Witchcraft:

  1. Read the TypeClass README
  2. Implement the TypeClass data generator protocol for your struct
  3. Use definst ("define instance") instead of defimpl:
definst Witchcraft.Functor, for: Algae.Id do
  def map(%{id: data}, fun), do: %Algae.Id{id: fun.(data)}

All classes have properties that your instance must conform to at compile time. mix will alert you to any failing properties by name, and will refuse to compile without them. Sometimes it is not possible to write an instance that will pass the check, and you can either write a custom generator for that instance, or force the instance. If you must resort to forcing the instance, please write a test of the property for some specific case to be reasonably sure that it will be compatible with the rest of the library.

More reference instances are available in Algae.


Family Function Operator
Setoid equivalent? ==
nonequivalent? !=
Ord greater_than? >
lesser_than? <
Semigroup append <>
Functor lift ~>
convey ~>>
chain >>>
over <~
ap <<~
reverse_chain <<<
Semigroupoid compose <|>
pipe_compose <~>
Arrow product ^^^
fanout &&&

Haskell Translation Table

Haskell Prelude Witchcraft
flip ($) |>/2 (Kernel)
. <|>/2
<<< <|>/2
>>> <~>/2
<> <>/2
<$> <~/2
flip (<$>) ~>/2
fmap lift/2
liftA lift/2
liftA2 lift/3
liftA3 lift/4
liftM lift/2
liftM2 lift/3
liftM3 lift/4
ap ap/2
<*> <<~/2
<**> ~>>/2
*> then/2
<* following/2
pure of/2
return of/2
>> then/2
>>= >>>/2
=<< <<</2
*** ^^^/2
&&& &&&/2

Prior Art and Further Reading

This library draws heavy inspiration from mathematics, other languages, and other Elixir libraries. We would be a shame not to mention them here. There is much, much more out there, but these are our highlights and inspirations.

The Monad library predates Witchcraft by several years. This library proved that it is entirely possible to bring do-notation to Elixir. It takers a very different approach: it is very up-front that it has a very loose definition of what it means for something to be a "monad", and relies on behaviours rather than ad-hoc polymorphism.

The Fantasy Land Spec is a spec for projects such as this one, but targeted at Javascript. It does not come with its own implementation, but provides a helpful chart of class hierarchies.

In many ways, Scalaz, and later cats, were the first widely-used port of categorical & algebraic ideas to a mainstream language. While dismissed by some as "Haskell fan fiction", it showed that we can write our own Haskell fanfic in all sorts of languages.

Obviously the Haskell Prelude deserves mention. Haskell has inspired so many programmers to write clean, declarative, functional code based on principled abstractions. We'll spare you the love letter to SPJ, the Glasgow team, and the original Haskell committee, but we're deeply appreciative of how they pushed the state of the art forward.

classy-prelude/mono-traversable have also made a lot of progress towards a base library that incorporates modern ideas in a clean package, and was an inspiration to taking a similar approach with Witchcraft.

The semigroupoids library from the eminent Edward Kmett provided many reference implementations and is helping set the future expansion of the foldable class lineage in Witchcraft.

Interested in learning more of the underlying ideas? The maintainers can heavily recommend Conceptual Mathematics, Category Theory for the Sciences, and Categories for the Working Mathematician. Reading these books probably won't change your code overnight. Some people call it "general abstract nonsense" for a reason. That said, it does provide a nice framework for thinking about these abstract ideas, and is a recommended pursuit for all that are curious.



A big thank you to Brandon Labbé for creating the project logo.


Robot Overlord sponsors much of the development of Witchcraft, and dogfoods the library in real-world applications.