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    <p>The definition "Monads are just monoids in the category of endofunctors.", which although true is a bad place to start. It's from a <a href="http://james-iry.blogspot.com/2009/05/brief-incomplete-and-mostly-wrong.html" rel="noreferrer">blog post</a> that was largely intended to be a joke. But if you are interested in the correspondence it can be demonstrated in Haskell:</p> <p>The laymen description of a category is an abstract collection of objects and morphisms between objects. Mappings between categories are called <em>functors</em> and map objects to objects and morphisms to morphisms associatively and preserves identities. An <em>endofunctor</em> is a functor from a category to itself.</p> <pre><code>{-# LANGUAGE MultiParamTypeClasses, ConstraintKinds, FlexibleInstances, FlexibleContexts #-} class Category c where id :: c x x (.) :: c y z -&gt; c x y -&gt; c x z class (Category c, Category d) =&gt; Functor c d t where fmap :: c a b -&gt; d (t a) (t b) type Endofunctor c f = Functor c c f </code></pre> <p>Mappings between functors which satisfy the so called <a href="https://en.wikipedia.org/wiki/Natural_transformation#Definition" rel="noreferrer">naturality conditions</a> are called <em>natural transformations</em>. In Haskell these are polymorphic functions of type: <code>(Functor f, Functor g) =&gt; forall a. f a -&gt; g a</code>.</p> <p>A <em>monad</em> on a category <code>C</code> is three things <code>(T,η,μ)</code>, <code>T</code> is endofunctor and <code>1</code> is the identity functor on <code>C</code>. Mu and eta are two natural transformations that satisfy a <a href="http://ncatlab.org/nlab/show/monad#definition_18" rel="noreferrer">triangle identity</a> and an associativity identity, and are defined as:</p> <ul> <li><code>η : 1 → T</code></li> <li><code>μ : T^2 → T</code></li> </ul> <p>In Haskell <code>μ</code> is <code>join</code> and <code>η</code> is <code>return</code> </p> <ul> <li><code>return :: Monad m =&gt; a -&gt; m a</code></li> <li><code>join :: Monad m =&gt; m (m a) -&gt; m a</code></li> </ul> <p>A categorical definition of Monad in Haskell could be written:</p> <pre><code>class (Endofunctor c t) =&gt; Monad c t where eta :: c a (t a) mu :: c (t (t a)) (t a) </code></pre> <p>The bind operator can be derived from these.</p> <pre><code>(&gt;&gt;=) :: (Monad c t) =&gt; c a (t b) -&gt; c (t a) (t b) (&gt;&gt;=) f = mu . fmap f </code></pre> <p>This is a complete definition, but equivalently you can also show that the Monad laws can be expressed as Monoid laws with a <em>functor category</em>. We can construct this functor category which is a category with objects as functors ( i.e. mappings between categories) and natural transformations (i.e. mappings between functors) as morphisms. In a category of endofunctors all functors are functors between the same category.</p> <pre><code>newtype CatFunctor c t a b = CatFunctor (c (t a) (t b)) </code></pre> <p>We can show this gives rise to a category with functor composition as morphism composition:</p> <pre><code>-- Note needs UndecidableInstances to typecheck instance (Endofunctor c t) =&gt; Category (CatFunctor c t) where id = CatFunctor id (CatFunctor g) . (CatFunctor f) = CatFunctor (g . f) </code></pre> <p>The monoid has the usual definition:</p> <pre><code>class Monoid m where unit :: m mult :: m -&gt; m -&gt; m </code></pre> <p>A monoid over a category of functors has a natural transformations as identity a and a multiplication operation which combines natural transformations. Kleisli composition can be defined to satisfy the multiplication law.</p> <pre><code>(&lt;=&lt;) :: (Monad c t) =&gt; c y (t z) -&gt; c x (t y) -&gt; c x (t z) f &lt;=&lt; g = mu . fmap f . g </code></pre> <p>And so you have it "Monads are just monoids in the category of endofunctors" which is just a "pointfree" version of the normal definition of monads from endofunctors and (mu, eta).</p> <pre><code>instance (Monad c t) =&gt; Monoid (c a (t a)) where unit = eta mult = (&lt;=&lt;) </code></pre> <p>With a bit of substitution one can show that the monoidal properties of <code>(&lt;=&lt;)</code> Are equivalent statement of the triangle and associativity diagrams for the monad's natural transformations.</p> <pre><code>f &lt;=&lt; unit == f unit &lt;=&lt; f == f f &lt;=&lt; (g &lt;=&lt; h) == (f &lt;=&lt; g) &lt;=&lt; h </code></pre> <p>If you're interested in <a href="http://www.stephendiehl.com/posts/monads.html" rel="noreferrer">diagrammatic representations</a> I have written a bit about representing them with string diagrams.</p>
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