\[ \require{mathtools} \let\DeclarePairedDelimiter\DeclarePairedDelimiters % MathJax typo % sup \newcommand{\op}{^\mathrm{op}} \newcommand{\inv}{^{-1}} % rm \newcommand{\compL}{\mathbin{\circ}} \newcommand{\Kl}{\mathrm{Kl}} % arrow \newcommand{\xto}{\xrightarrow} \newcommand{\xot}{\xleftarrow} \newcommand{\toot}{\rightleftarrows} \newcommand{\klto}{\rightsquigarrow} \newcommand{\comma}{\downarrow} \newcommand{\incl}{\hookrightarrow} \newcommand{\mono}{\rightarrowtail} \newcommand{\epi}{\twoheadrightarrow} \newcommand{\iso}{\cong} \newcommand{\nat}{\Rightarrow} \newcommand{\adj}{\dashv} % delimiter \DeclarePairedDelimiter{\parens}{(}{)} % parentheses ( ) \DeclarePairedDelimiter{\bracks}{[}{]} % brackets [ ] \DeclarePairedDelimiter{\braces}{\{}{\}} % braces { } \DeclarePairedDelimiter{\angles}{\langle}{\rangle} % angles \newcommand{\set}{\braces} % operator \DeclareMathOperator{\Obj}{Obj} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Eq}{Eq} \DeclareMathOperator{\Coeq}{Coeq} \DeclareMathOperator{\Pull}{Pull} \DeclareMathOperator{\Push}{Push} \DeclareMathOperator{\Lan}{Lan} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\Lift}{Lift} \DeclareMathOperator{\Rift}{Rift} % \DeclareMathOperator{\id}{id} \DeclareMathOperator{\act}{act} \DeclareMathOperator{\colim}{colim} % exponential \DeclareMathOperator{\partf}{partial} \DeclareMathOperator{\curry}{curry} % natural number \DeclareMathOperator{\zero}{zero} \DeclareMathOperator{\successor}{succ} % list \DeclareMathOperator{\nil}{nil} \DeclareMathOperator{\cons}{cons} \newcommand{\concat}{\mathbin{{+}\mspace{-8mu}{+}}} % fold \DeclareMathOperator{\fold}{fold} \DeclareMathOperator{\map}{map} \DeclareMathOperator{\filter}{filter} % bool \DeclareMathOperator{\cond}{cond} % cd \DeclareMathOperator{\copy}{copy} \DeclareMathOperator{\del}{del} % sets \newcommand{\N}{\mathbb{N}} % natural numbers \newcommand{\Z}{\mathbb{Z}} % integers \newcommand{\Q}{\mathbb{Q}} % rational numbers \newcommand{\R}{\mathbb{R}} % real numbers % category \newcommand{\catf}[1]{{\mathbf{#1}}} \newcommand{\cA}{\catf{A}} \newcommand{\cB}{\catf{B}} \newcommand{\cC}{\catf{C}} \newcommand{\cD}{\catf{D}} \newcommand{\cE}{\catf{E}} \newcommand{\cI}{\catf{I}} \newcommand{\cJ}{\catf{J}} \newcommand{\cS}{\catf{S}} \newcommand{\cT}{\catf{T}} \newcommand{\cV}{\catf{V}} \newcommand{\cZero}{\catf{0}} \newcommand{\cOne}{\catf{1}} \newcommand{\cTwo}{\catf{2}} % \newcommand{\cArr}{\catf{Arr}} % arrow \newcommand{\cPSh}{\catf{PSh}} % presheaf \newcommand{\cCone}{\catf{Cone}} % cone \newcommand{\cCocone}{\catf{Cocone}} % cocone % \newcommand{\cFin}{\catf{Fin}} % finite prefix \newcommand{\cSet}{\catf{Set}} % functions \newcommand{\cRel}{\catf{Rel}} % relations \newcommand{\cPos}{\catf{Pos}} % posets \newcommand{\cMon}{\catf{Mon}} % monoids \newcommand{\cGrp}{\catf{Grp}} % groups \newcommand{\cAb}{\catf{Ab}} % abelian groups \newcommand{\cCat}{\catf{Cat}} % categories \newcommand{\cGrpd}{\catf{Grpd}} % groupoids \newcommand{\cVect}{\catf{Vect}} % vector spaces \newcommand{\cMeas}{\catf{Meas}} % measurable spaces and measurable functions \newcommand{\cStoch}{\catf{Stoch}} %measurable spaces and stochastic maps \newcommand{\cProb}{\catf{Prob}} % probability measures and measure-preserving functions % logic \newcommand{\lfal}{\bot} \newcommand{\ltru}{\top} \]

A monad is a structure that describes generalized objects and morphisms.

1 Monad

The category \([\cC, \cC]\) of endofunctors is a strict monoidal category with the functor composition as the monoidal multiplication and the identity functor \(\id_\cC\) as the monoidal unit.

A monad in a category \(\cC\) is an endofunctor \(T: \cC \to \cC\) equipped with two natural transformations:

composition \(\mu: TT \nat T\)
unit \(\eta: \id_\cC \nat T\)

subject to

associativity: \(\mu (T \mu) = \mu (\mu T)\)
identity: \(\mu (T \eta) = \mu (\eta T) = \id_T\)

In other words, a monad is a monoid in the category of endofunctors.

-- bind
(>>=) :: Monad m => m a -> (a -> m b) -> m b
(=<<) :: Monad m => (a -> m b) -> m a -> m b
(>>)  :: Monad m => m a -> m b -> m b
(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> (a -> m c)
(<=<) :: Monad m => (b -> m c) -> (a -> m b) -> (a -> m c)

2 T-algebra

For a monad \((T, \mu, \eta)\), a (left) algebra over a monad (T-algebra) is an M-algebra \((A, \act_A: TA \to A)\) in the category of endofunctors over the constant functor \(\Delta A\).

Thus, a T-algebra is compatible with

monad composition: \(\act_A \compL T \act_A = \act_A \compL \mu_A\)
monad unit: \(\act_A \compL \eta_A = \id_A\)

A T-algebra homomorphism is an F-algebra homomorphism.

3 Eilenberg–Moore category

For a monad \((T, \mu, \eta)\) in a category \(\cC\), the Eilenberg–Moore category \(\cC^T\) is a category of T-algebras and T-algebra homomorphisms.

4 Kleisli category

For a monad \((T, \mu, \eta)\) in a category \(\cC\), the Kleisli category \(\cC_T\) is a category whose

objects are \(\cC\)-objects
morphisms are Kleisli morphisms \(A \xto{f_\Kl} TB\)
composition is the Kleisli composition \(g_\Kl \compL_\Kl f_\Kl := \mu \compL Tg_\Kl \compL f_\Kl\)

5 Monadic adjunction

5.1 Adjunction to monad

An adjunction

\[(F: \cC \toot \cD: G, \epsilon: FG \nat \id_\cD, \eta: \id_\cC \nat GF)\]

gives rise to a monad

\[(GF: \cC \to \cC, G \epsilon F: GFGF \nat GF, \eta: \id_\cC \nat GF)\]

and a comonad

\[(FG: \cD \to \cD, F \eta G: FG \nat FGFG, \epsilon: FG \nat \id_\cD)\]

5.2 Monad to adjunction

6 Monad examples

6.1 Maybe

\(MA = 1 + A\) nothing or values
\(Mf = 1 + f\) computation with failure
\(\mu: MM \nat M\) failure propagation
\(\eta: \id \nat M\) just
\(A \to MB\) partial function
\(MA \to A\) pointed object

6.2 List

free monoid

\(\nil: 1 \to LA\) empty list
\(\concat: LA \times LA \to LA\) concatenation

free monoid monad

\(LA\) list
\(Lf\) list map
\(\mu: LL \nat L\) concatenation
\(\eta: \id \nat L\) singleton
\(A \to LB\) list function
\(LA \to A\) monoid \((A, \fold [e: 1 \to A, \cdot: A \times A \to A])\)
- right identity \([a]\)
- left identity \([[], [a]]\)
- associativity \([[a, b], [c]]\)

6.3 Powerset

\(PA = 2^A\) powerset
\(Pf\) set map
\(\mu: PP \nat P\) union
\(\eta: \id \nat P\) singleton
\(A \to PB\) set function
\(PA \to A\) suplattice

6.4 Multiset

\(NA = \N^A\) multiset
\(Nf\) sum map
\(\mu: NN \nat N\) multiplication
\(\eta: \id \nat N\) singleton
\(A \to NB\) multiset function
\(NA \to A\) commutative monoid

6.5 Probability

\(PA\) probability measures
\(Pf\) pushforward measure
\(\mu: PP \nat P\) marginalization
\(\eta: \id \nat P\) Dirac measure
\(A \to PB\) conditional probability
\(PA \to A\) expectation

6.6 State

\([S, S \times A]\)

6.7 Continuation

\([[A, R], R]\)

7 Monoidal monad

A monoidal monad is a monad in the 2-category of monoidal categories, lax monoidal functors, and monoidal natural transformations.

8 Monoidal Kleisli category

For a monoidal monad \((T, \mu_T, \eta_T, \mu, \eta)\) in \((\cC, \otimes, I)\), the Kleisli category \(\cC_T\) inherits the monoidal structure from \(\cC\), whose

monoidal multiplication \(\otimes_\Kl\) on objects is the monoidal multiplication \(\otimes\) on \(\cC\)
monoidal multiplication \(\otimes_\Kl\) on morphisms is obtained by \(f_\Kl \otimes_\Kl g_\Kl := \mu (f_\Kl \otimes g_\Kl)\)

\[(A \xto{f_\Kl} TB) \otimes_\Kl (C \xto{g_\Kl} TD) = A \otimes C \xto{f_\Kl \otimes_\Kl g_\Kl} T(B\otimes D) = A \otimes C \xto{f_\Kl \otimes g_\Kl} TB \otimes TD \xto{\mu_{B, D}} T(B \otimes D)\]
monoidal unit, associator, and left/right unitors are those of \(\cC\)