diff --git a/src/algebra/free.js b/src/algebra/free.js
index 9bf132d..536ed01 100644
--- a/src/algebra/free.js
+++ b/src/algebra/free.js
@@ -1,145 +1,145 @@
-import { id, map, ap, prepend, reduce, thunk } from '../../vendor/izuna/src/index.js'
-/** @import { Morphism } from './types.js' */
+import { Algebra, Monad } from './interfaces.js'
+import { curry } from '../..//vendor/izuna/src/index.js'
 
+/** @import { Applicative, ApplicativeTypeRef, Morphism } from './types.js' */
+
+
+/** @import { InferredMorphism } from './types.js' */
+
+const kleisli = curry((f, g, x) => f(x).chain(g))
 
 /**
- * @template A
- * @typedef {Pure<A> | Impure<A>} Free
+ * @template T
+ * @type {Pure<T> | Impure<T>} Free
  */
+export class Free extends Algebra(Monad) {
+  /**
+   * @template T
+   * @param {T} value 
+   */
+  static of(value) {
+    return new Pure(value)
+  }
+}
 
-/** @template A */
-export class Pure {
+/** @template T */
+export class Pure extends Free {
   #value
 
   /**
-   * @param {A} value 
+   * @param {T} value 
    */
   constructor(value) {
+    super()
     this.#value = value
   }
 
   /**
-   * @template {Free<B>} B
-   * @param {Morphism<A, B>} f
-   * @returns {B}
+   * @param {InferredMorphism<T>} f 
    */
   chain(f) {
     return f(this.#value)
   }
 
   /**
-   * @template B
-   * @param {Morphism<A, B>} f
-   * @returns {Free<B>}
+   * @param {InferredMorphism<T>} f 
    */
   map(f) {
-    // @ts-ignore
     return pure(f(this.#value))
   }
 
   /**
-   * @template B, C
-   * @param {Free<B>} b 
-   * @returns {Free<C>}
+   * @template U
+   * @template {Applicative<U>} M
+   * @template {ApplicativeTypeRef<U, M>} TypeRef
+   * @param {TypeRef} _A
+   * @param {*} f
+   * @returns {M}
    */
-  ap(b) {
-    return b.map(this.#value)
+  traverse(_A, f) {
+    return f(this.#value).map(pure)
   }
 
-  traverse(f, of) {
-    return of(this.map(f))
+  run() {
+    return this.#value
   }
-
-  /**
-   * @template B
-   * @param {(acc: B, value: A) => B} f 
-   * @param {B} init
-   * @returns {B}
-   */
-  reduce(f, init) {
-    return f(init, this.#value)
-  }
-
 }
 
-/**
- * @template A
- * @typedef {<B>(value: A) => Free<B>} Computation
- */
-
-/** @template A */
-export class Impure {
+/** @template T */
+export class Impure extends Free {
+  #value
   #next
 
   /**
-   * @param {() => Free<any>} next 
+   * @param {T} value 
+   * @param {(value: T) => Free<any>} f 
    */
-  constructor(next) {
-    this.#next = next
+  constructor(value, f) {
+    super()
+    this.#value = value
+    this.#next = f
   }
 
   /**
-   * @template {Free<B>} B
-   * @param {Morphism<A, B>} f
-   * @returns {B}
+   * @param {<U>(value: T) => Free<U>} f 
    */
   chain(f) {
-    // @ts-ignore
-    return new Impure(
-      // @ts-ignore
-      () => this.#next().chain(f)
-    )
+    return impure(this.#value, kleisli(this.#next, f))
   }
 
-  /**
-   * @template B
-   * @param {Morphism<A, B>} f
-   * @returns {Free<B>}
-   */
   map(f) {
-    return new Impure(
-      // @ts-ignore
-      () => this.#next().map(f)
+    return impure(
+      this.#value,
+      y => f(y).chain(f)
     )
   }
 
   /**
-   * @template B
-   * @param {Free<any>} b 
-   * @returns {Free<B>}
+   * @template U
+   * @template {Applicative<U>} M
+   * @template {ApplicativeTypeRef<U, M>} TypeRef
+   * @param {TypeRef} A
+   * @param {(value: T) => M} f
+   * @returns {M}
    */
-  ap(b) {
-    return new Impure(
-      // @ts-ignore
-      () => b.map(this.#next())
+  traverse(A, f) {
+    const fb = f(this.#value)
+    const rest = this.#next(this.#value).traverse(A, f)
+
+    return rest.ap(
+      fb.map(b => next => impure(b, () => next))
     )
   }
 
-  traverse(f, of) {
-    return of(liftF(
-      this.#next().traverse(f, of)
-    ))
-  }
-
-  /**
-   * @template B
-   * @param {(acc: B, value: A) => B} f 
-   * @param {B} acc
-   * @returns {B}
-   */
-  reduce(f, acc) {
-
+  run() {
+    return this.#next(this.#value).run()
   }
 }
 
-const sequence = (of, iter) => {
-  return reduce((acc, x) => {
-    ap(acc, map(prepend, x))
-  }, of([]), iter)
-}
+/**
+ * @template T
+ * @param {T} x
+ * @returns {Pure<T>}
+ */
+const pure = x => new Pure(x)
 
-export const pure = x => new Pure(x)
-export const impure = f => new Impure(f)
+/**
+ * @template T
+ * @param {T} x
+ * @param {(value: T) => Free<any>} f
+ * @returns {Impure<T>}
+ */
+const impure = (x, f) => new Impure(x, f)
 
-export const liftF = effect => impure(thunk(effect))
+/**
+ * @template T
+ * @param {T} x
+ * @returns Impure<T>
+ */
+export const liftF = x => impure(x, pure)
+import { id, Identity } from './identity.js'
+import { Option, some } from './option.js'
+
+const a = liftF(1).chain(() => liftF(2)).traverse(Option, x => some(1))
+console.log(a.chain(x => x).run())
 
diff --git a/src/algebra/identity.js b/src/algebra/identity.js
new file mode 100644
index 0000000..d892b41
--- /dev/null
+++ b/src/algebra/identity.js
@@ -0,0 +1,65 @@
+import { Algebra, Comonad, Monad } from './interfaces.js'
+
+/** @import { Morphism } from './types.js' */
+
+/** @template T */
+export class Identity extends Algebra(Monad, Comonad) {
+  #value
+
+  /**
+   * @param {T} value 
+   */
+  constructor(value) {
+    super()
+
+    this.#value = value
+  }
+
+  /**
+   * @template T
+   * @param {T} value 
+   */
+  static of(value) {
+    return id(value)
+  }
+
+  /**
+   * @template U
+   * @param {Morphism<T, U>} f 
+   */
+  map(f) {
+    return id(f(this.#value))
+  }
+
+  /**
+   * @template U
+   * @this {Identity<Morphism<U, T>>}
+   * @param {Identity<U>} b 
+   * @returns {Identity<T>}
+   */
+  ap(b) {
+    return b.map(this.#value)
+  }
+
+  /**
+   * @template U
+   * @param {Morphism<T, Identity<U>>} f 
+   */
+  chain(f) {
+    return f(this.#value)
+  }
+
+  /**
+   * @param {(value: Identity<T>) => T} f 
+   */
+  extend(f) {
+    return id(f(this))
+  }
+
+  extract() {
+    return this.#value
+  }
+}
+
+export const id = value => new Identity(value)
+
diff --git a/src/algebra/interfaces.js b/src/algebra/interfaces.js
index 08b6b9d..f877198 100644
--- a/src/algebra/interfaces.js
+++ b/src/algebra/interfaces.js
@@ -1,4 +1,4 @@
-import { mix } from '../mixin.js'
+import { mix, Mixin } from '../mixin.js'
 
 /** @import { MixinFunction } from '../mixin.js' */
 /** @import { Fn } from './types.js' */
@@ -130,7 +130,7 @@ class Method {
    */
   _getInstallationPoint(target) {
     switch (this.#type) {
-      case 0: return target.prototype
+      case 0: return Object.getPrototypeOf(target)
       case 1: return target
       default: return target
     }
@@ -152,6 +152,9 @@ const Implementations = Symbol()
 class Interface {
   #name
 
+  /** @type {MixinFunction} */
+  #mixin
+
   /** @type {Set<Method>} */
   #methods = new Set()
 
@@ -161,20 +164,20 @@ class Interface {
   /** @param {string} name */
   constructor(name) {
     this.#name = name
+    this.#mixin = Mixin(this.build.bind(this))
   }
 
   get name() {
     return this.#name
   }
 
-  findInterfaces() {
+  findInterfaces(type) {
     let interfaces = new Set()
-    let current = Object.getPrototypeOf(this)
-    console.log(current)
+    let current = type
 
     while (current != null) {
-      interfaces = new Set(current[Implementations])
-      current = Object.getPrototypeOf(this)
+      interfaces = new Set(current[Implementations]).union(interfaces)
+      current = Object.getPrototypeOf(current)
     }
 
     return interfaces
@@ -185,7 +188,7 @@ class Interface {
    * @returns {boolean}
    */
   implementedBy(type) {
-    return this.findInterfaces().has(type)
+    return this.findInterfaces(type).has(this)
   }
 
   /**
@@ -212,10 +215,10 @@ class Interface {
   }
 
   /**
-   * @returns {MixinWrapper}
+   * @returns {MixinFunction}
    */
-  intoWrapper() {
-    return this.build.bind(this)
+  asMixin() {
+    return this.#mixin
   }
 
   /**
@@ -223,19 +226,20 @@ class Interface {
    */
   build(base) {
     const interfaces = [...this.#interfaces.values()]
-    const wrappers = interfaces.map(x => x.intoWrapper())
+    const mixins = interfaces.map(x => x.asMixin())
 
-    const Interfaces = mix(base).with(...wrappers)
+    const Interfaces = mix(base).with(...mixins)
 
-    //const Algebra = class extends Interfaces { }
+    const Algebra = class extends Interfaces { }
 
     for (const method of this.#methods) {
-      method.implement(this, Interfaces)
+      method.implement(this, Algebra)
     }
 
-    Interfaces.prototype[Implementations] = new Set([...interfaces])
+    const prototype = Object.getPrototypeOf(Algebra)
+    prototype[Implementations] = new Set(interfaces)
 
-    return Interfaces
+    return Algebra
   }
 }
 
@@ -272,7 +276,7 @@ export const apply = (methodName, f, obj) => {
  * @returns {(...algebras: Interface[]) => FunctionConstructor}
  */
 export const AlgebraWithBase = base => (...algebras) => {
-  return mix(base).with(...algebras.map(x => x.intoWrapper()))
+  return mix(base).with(...algebras.map(x => x.asMixin()))
 }
 
 /**
diff --git a/src/algebra/option.js b/src/algebra/option.js
index 945431d..60d8a21 100644
--- a/src/algebra/option.js
+++ b/src/algebra/option.js
@@ -2,8 +2,22 @@ import { Algebra, Foldable, Monad, Monoid, Setoid } from './interfaces.js'
 
 /** @import { Morphism } from './types.js' */
 
+/**
+ * @template T
+ * @type {Some<T> | None<T>} Option
+ */
+export class Option extends Algebra(Setoid, Monad, Foldable) {
+  /**
+   * @template T
+   * @param {T} value
+   */
+  static of(value) {
+    return Some.of(value)
+  }
+}
+
 /** @template T */
-export class Some extends Algebra(Setoid, Monoid, Monad, Foldable) {
+export class Some extends Option {
   /** @type {T} */
   #value
 
@@ -23,11 +37,18 @@ export class Some extends Algebra(Setoid, Monoid, Monad, Foldable) {
     return new Some(value)
   }
 
+  /**
+   * @param {Option<T>} other 
+   */
   equals(other) {
     return other instanceof Some &&
       other.chain(v => v === this.#value)
   }
 
+  ap(b) {
+    return b.chain(f => this.map(f))
+  }
+
   /**
    * @template R
    * @param {Morphism<T, R>} f
@@ -64,18 +85,21 @@ export class Some extends Algebra(Setoid, Monoid, Monad, Foldable) {
   reduce(f, init) {
     return f(init, this.#value)
   }
-
-  empty() {
-    return Option.empty()
-  }
 }
 
 /** @template T */
-export class None extends Algebra(Setoid, Monoid, Monad, Foldable) {
+export class None extends Option {
+  /**
+   * @param {Option<T>} other 
+   */
   equals(other) {
     return other === none
   }
 
+  ap(_b) {
+    return this
+  }
+
   /**
    * @template R
    * @param {Morphism<T, R>} _f
@@ -118,24 +142,6 @@ export class None extends Algebra(Setoid, Monoid, Monad, Foldable) {
   }
 }
 
-/**
- * @template T
- * @type {Some<T> | None<T>} Option
- */
-export class Option {
-  /**
-   * @template T
-   * @param {T} value
-   */
-  static of(value) {
-    return Some.of(value)
-  }
-
-  static empty() {
-    return none
-  }
-}
-
 /**
  * @template T
  * @param {T} value
diff --git a/src/algebra/result.js b/src/algebra/result.js
index aef2458..da59344 100644
--- a/src/algebra/result.js
+++ b/src/algebra/result.js
@@ -1,11 +1,26 @@
-import { Algebra, Foldable, Functor, Monad, Setoid } from './interfaces.js'
+import { Algebra, Foldable, Monad, Setoid } from './interfaces.js'
 
 /** @import { Morphism } from './types.js' */
 
 /**
  * @template T, E
+ * @type {Ok<T, E> | Err<T, E>} Result
  */
-export class Ok extends Algebra(Setoid, Monad, Foldable) {
+export class Result extends Algebra(Setoid, Monad, Foldable) {
+  /**
+   * @template T
+   * @param {T} value 
+   * @returns {Ok}
+   */
+  static of(value) {
+    return new Ok(value)
+  }
+}
+
+/**
+ * @template T, E
+ */
+export class Ok extends Result {
   /** @type {T} */
   #value
 
@@ -20,7 +35,7 @@ export class Ok extends Algebra(Setoid, Monad, Foldable) {
   }
 
   /**
-   * @param {Ok<T, E>} other 
+   * @param {Result<T, E>} other 
    */
   equals(other) {
     return other instanceof Ok &&
@@ -86,7 +101,7 @@ export class Ok extends Algebra(Setoid, Monad, Foldable) {
 /**
  * @template T, E
  */
-export class Err extends Algebra(Setoid, Functor, Monad) {
+export class Err extends Result {
   /** @type {E} */
   #value
 
@@ -100,6 +115,9 @@ export class Err extends Algebra(Setoid, Functor, Monad) {
     this.#value = value
   }
 
+  /**
+   * @param {Result<T, E>} other 
+   */
   equals(other) {
     return other instanceof Err &&
       other.chain(v => v === this.#value)
@@ -158,21 +176,6 @@ export class Err extends Algebra(Setoid, Functor, Monad) {
   }
 }
 
-/**
- * @template T, E
- * @type {Ok<T, E> | Err<T, E>} Result
- */
-export class Result {
-  /**
-   * @template T
-   * @param {T} value 
-   * @returns {Ok}
-   */
-  static of(value) {
-    return new Ok(value)
-  }
-}
-
 /**
  * @template T, E
  * @param {T} v
diff --git a/src/algebra/types.js b/src/algebra/types.js
index a3690c7..d8b95d6 100644
--- a/src/algebra/types.js
+++ b/src/algebra/types.js
@@ -17,17 +17,197 @@ export default {}
 
 /**
  * @template T
- * @typedef {(value: T) => Chain<T>} ChainConstructor
+ * @typedef {{
+     equals: (b: Setoid<T>) => boolean
+ * }} Setoid
  */
 
 /**
  * @template T
- * @typedef {<R>(f: Morphism<T, R>) => R} chain
+ * @typedef {{
+     lte: (b: Ord<T>) => boolean
+ * }} Ord
+ */
+
+/**
+ * @template T, U
+ * @typedef {*} Semigroupoid
+ */
+
+/**
+ * @template I, J
+ * @typedef {{
+     compose: <K>(b: Semigroupoid<J, K>) => Semigroupoid<I, K>
+ * }} Category
+ */
+
+/**
+ * @template T, U
+ * @template {Category<T, U>} M
+ * @typedef {{
+    id: () => (value: T) => M
+ * }} CategoryTypeRef
  */
 
 /**
  * @template T
- * @typedef {{ chain: chain<T> }} Chain
+ * @typedef {{
+     concat: (b: Semigroup<T>) => Semigroup<T>
+ * }} Semigroup
+ */
+
+/**
+ * @template T
+ * @typedef{Semigroup<T>} Monoid
+ */
+
+/**
+ * @template T
+ * @template {Monoid<T>} M
+ * @typedef {{
+     empty: () => M
+ * }} MonoidTypeRef
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Monoid<T> &
+     { invert: () => Group<T> }
+ * } Group
+ */
+
+/**
+ * @template T
+ * @typedef {{
+     filter: <U>(f: (acc: U, val: T) => U, init: U) => U
+ * }} Filterable
+ */
+
+/**
+ * @template T
+ * @typedef {{
+     map: <U>(f: Morphism<T, U>) => Functor<U>
+ * }} Functor
+ */
+
+/**
+ * @template T
+ * @typedef {{
+     contramap: <U>(f: Morphism<U, T>) => Contravariant<T>
+ * }} Contravariant
+ */
+
+/**
+ * @template T
+ * @typedef {{
+     ap: <U>(f: Apply<Morphism<T, U>>) => Apply<U>
+ * }} Apply
+ */
+
+/**
+ * @template T
+ * @typedef {{
+    ap: <U>(b: Applicative<Morphism<U, T>>) => Applicative<T>,
+    map: <U>(f: Morphism<T, U>) => Applicative<U>
+ * }} Applicative
+ */
+
+/**
+ * @template T
+ * @template {Applicative<T>} M
+ * @typedef {{
+     of: (value: T) => M
+ * }} ApplicativeTypeRef
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Functor<T> &
+     { alt: (b: Alt<T>) => Alt<T> }
+ * } Alt
+ */
+
+/**
+ * @template T
+ * @typedef {Alt<T>} Plus
+ */
+
+/**
+ * @template T
+ * @template {Plus<T>} M
+ * @typedef {
+     Alt<T> &
+     { zero: () => M }
+ * } PlusTypeRef
+ */
+
+/**
+ * @template T
+ * @typedef {Applicative<T> & Plus<T>} Alternative
+ */
+
+/**
+ * @template T
+ * @template {Applicative<T> & Plus<T>} M
+ * @typedef {ApplicativeTypeRef<T, M> & PlusTypeRef<T, M>} AlternativeTypeRef
+ */
+
+/**
+ * @template T
+ * @typedef {{
+     filter: <U>(f: (acc: U, val: T) => U, init: U) => U
+ * }} Foldable
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Functor<T> & Foldable<T> &
+     { traverse: <U>(A: ApplicativeTypeRef<U, Applicative<U>>, f: (val: T) => Applicative<U>) => Applicative<Traversable<U>> }
+ * } Traversable
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Apply<T> &
+     { chain: <U>(f: (value: T) => Chain<U>) => Chain<U> }
+ * } Chain
+ */
+
+/**
+ * @template T
+ * @typedef {Chain<T>} ChainRec
+ */
+
+/**
+ * @template T
+ * @typedef {Functor<T> & Applicative<T> & Chain<T>} Monad
+ */
+
+/**
+ * @template T
+ * @template {Monad<T>} M
+ * @typedef {ApplicativeTypeRef<T, M>} MonadTypeDef
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Functor<T> &
+     { extend: <U>(f: (val: Extend<T>) => U) => Extend<U>}
+ * } Extend
+ */
+
+/**
+ * @template T
+ * @typedef {
+     Extend<T> &
+     { extract: () => T }
+ * } Comonad<T>
  */
 
 /** @typedef {(...args: any[]) => any} Fn */
+
diff --git a/src/mixin.js b/src/mixin.js
index 89fbe27..31fe6ad 100644
--- a/src/mixin.js
+++ b/src/mixin.js
@@ -246,7 +246,7 @@ class MixinBuilder {
   /**
    * Applies `mixins` in order to the superclass given to `mix()`.
    *
-   * @param {Array.<Mixin>} mixins
+   * @param {Array.<MixinFunction>} mixins
    * @return {FunctionConstructor} a subclass of `superclass` with `mixins` applied
    */
   with(...mixins) {
diff --git a/tests/units/monad.js b/tests/units/monad.js
index 6ce7ed4..c335d5f 100644
--- a/tests/units/monad.js
+++ b/tests/units/monad.js
@@ -1,28 +1,29 @@
 // @ts-nocheck
 import { it, assert } from 'folktest'
-import { Option, Result } from '../../src/index.js'
-import { Chain, Ord, Setoid } from '../../src/algebra/interfaces.js'
+import { IO, List, Option, Reader, Result, Some } from '../../src/index.js'
+import { Applicative, Chain, Ord, Setoid } from '../../src/algebra/interfaces.js'
 
-const Algebra = [Option, Result]
+const Algebra = [Option, Result, List, IO, Reader]
 
-const inc = x => F.of(x + 1)
-const dbl = x => F.of(x * 2)
+const inc = F => x => F.of(x + 1)
+const dbl = F => x => F.of(x * 2)
 
 const impl = i => m => i.implementedBy(m)
 
 export const Tests = [
   it('unit is a left identity', () => {
-    console.log(Chain.implementedBy(Result))
-    Algebra.filter(impl(Chain)).forEach(algebra => {
+    Algebra.filter(impl(Setoid, Applicative, Chain)).forEach(algebra => {
+      const f = inc(algebra)
+
       assert(
-        algebra.of(1).chain(inc) == inc(1),
+        algebra.of(1).chain(f).equals(f(1)),
         `${algebra.name} is not a left identity`
       )
     })
   }),
 
   it('unit is a right identity', () => {
-    Algebra.filter(impl(Chain)).forEach(algebra => {
+    Algebra.filter(impl(Setoid, Applicative, Chain)).forEach(algebra => {
       const m = algebra.of(1)
 
       assert(
@@ -33,36 +34,15 @@ export const Tests = [
   }),
 
   it('unit is associative', () => {
-    Algebra.filter(impl(Chain)).forEach(algebra => {
+    Algebra.filter(impl(Setoid, Applicative, Chain)).forEach(algebra => {
       const a = algebra.of(1)
+      const f = inc(algebra)
+      const g = dbl(algebra)
 
-      const x = a.chain(inc).chain(dbl)
-      const y = a.chain(x => inc(x).chain(dbl))
+      const x = a.chain(f).chain(g)
+      const y = a.chain(x => f(x).chain(g))
 
-      assert(x == y, `${algebra.name} is not associative`)
+      assert(x.equals(y), `${algebra.name} is not associative`)
     })
   }),
-
-  it('unit is reflexive', () => {
-    Algebra.filter(impl(Setoid)).forEach(algebra => {
-      const a = algebra.of(1)
-      assert(a.equals(a), `${algebra.name} is not reflexive`)
-    })
-  }),
-
-  it('unit is symmetrical', () => {
-    Algebra.filter(impl(Setoid)).forEach(algebra => {
-      const a = algebra.of(1)
-      const b = algebra.of(1)
-      assert(a.equals(b) && b.equals(a), `${algebra.name} is not symmetrical`)
-    })
-  }),
-
-  it('unit is antisymmetrical', () => {
-    Algebra.filter(impl(Ord)).forEach(algebra => {
-      const a = algebra.of(1)
-      const b = algebra.of(1)
-      assert(a.lte(b) && b.lte(a) && a.equals(b))
-    })
-  })
 ]