Monad

A monad is an endofunctor $T$ with unit and multiplication satisfying associativity and unit laws.

The unit repeats an element and multiplication selects the two outer diagonal entries. Both unit laws hold directly, and the associativity law selects the same two diagonal entries of an eight-entry array. Its algebras for a monad are precisely rectangular bands, including the empty algebra.

In a category with equalizers, the equalizer of ${\eta }_{TA},T{\eta }_A$ defines a functor $R$ by ${\beta }_{B}Rf=Tf{\beta }_A$. It has unit $\alpha$ and multiplication $\theta$ characterized byCancelling the monomorphisms ${\beta }_A$ reduces its unit and associativity laws to those of $T$, so $\beta :R\Rightarrow T$ is a monad morphism. The split equalizers associated with a monad show that ${\alpha }_{TA}$ is invertible. If every $T{\beta }_A$ is monic, then every $T{\alpha }_A$ is invertible, the naturality square of $\beta$ at ${\beta }_A$ is a pullback in a category, and $R$ is an idempotent monad. This monicity hypothesis is essential to this last argument; arbitrary functors need not preserve monomorphisms.

A morphism of monads on one category is a natural transformation $\alpha :S\Rightarrow T$ with $\alpha {\eta }^S={\eta }^T$ and $\alpha {\mu }^S={\mu }^T(T\alpha )(\alpha S)$. These equations preserve the unit and multiplication. A terminal endofunctor in a full composition-closed subcategory containing the identity receives a unique such morphism from every monad whose functor belongs to that subcategory.

A monad morphism $\alpha :S\Rightarrow T$ induces a functor ${\mathcal{C}}^T\to {\mathcal{C}}^S$ between Eilenberg-Moore categories, identical on underlying objects and arrows. Its unit equation gives the restricted algebra's unit law. Naturality and the multiplication equation give $(a{\alpha }_A)S(a{\alpha }_A)=aTa{\alpha }_{TA}S{\alpha }_A=a{\mu }_A^{T}T{\alpha }_A{\alpha }_{SA}=a{\alpha }_A{\mu }_A^S$. Naturality also preserves algebra morphisms.

On the Category of sets, the list monad sends $X$ to all finite ordered lists of elements of $X$, including the empty list. Its unit inserts a singleton list and its multiplication concatenates a list of lists. Functoriality is entrywise application of functions. Ordered flattening proves the monad laws. This is the free-monoid monad, so order must not be discarded.

An algebra for a monad for the list monad is precisely a monoid. For an action $a$, the identity is $a([])$ and multiplication is $a([x,y])$. The algebra laws force the unit and associativity laws and determine $a$ as ordered multiplication. Conversely ordered multiplication defines a list action. Algebra morphisms are exactly monoid homomorphisms, giving an isomorphism of categories ${\mathbf{Set}}^{\mathrm{List}}\cong \mathbf{Mon}$.

The natural maps $\eta :1\Rightarrow T$ and $\mu :T^2\Rightarrow T$ defining a monad, subject to the associativity and two unit identities.

A monad whose endofunctor is an opmonoidal functor and whose unit and multiplication are opmonoidal natural transformations. Its Eilenberg-Moore category has a tensor of algebras $(X,a)\otimes (Y,b)=(X\otimes Y,(a\otimes b)T_2)$ and unit $(I,T_0)$.

Let $M$ be the monoid of order-preserving functions $\mathbb{N}\to \mathbb{N}$, with $\mathbb{N}=\{0,1,\dots \}$, regarded as a one-object category. The endofunctor $T$ fixes that object and sends $f$ to $Tf(0)=0$, $Tf(n+1)=f(n)+1$. The displayed maps give its monad unit and multiplication. The multiplication is not injective, so this is not an idempotent monad. Its only algebra for a monad is $\mu$, because $a\eta =1$ forces $a(n+1)=n$ and monotonicity forces $a(0)=0$. Algebra endomorphisms are exactly the functions fixing zero. The Kleisli comparison functor sends $f$ to the function which is zero at zero and equals $f(n)$ at $n+1$; its inverse sends $h$ to $n\mapsto h(n+1)$. Thus the comparison is an isomorphism of categories even though the monad is not idempotent.

A monad is idempotent when its multiplication is invertible. Then $T\eta =\eta T={\mu }^{-1}$. If $(A,a)$ is an algebra for a monad, its unit law and naturality give $a{\eta }_A=1_A$ and ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1_{TA}$. Thus every algebra is isomorphic to a free algebra, and the Kleisli comparison functor into the Eilenberg-Moore category is an equivalence.

Let a full subcategory of the endofunctor category contain the identity, be closed under composition, and have terminal object $T$. Then $T$ has a unique monad structure: the unit $1\to T$ and multiplication $T^2\to T$ are the unique maps to the terminal object. The monad laws hold because each compares two maps with codomain $T$ from the same object.

The ultrafilter functor carries the monad structure obtained from its terminality among finite-coproduct-preserving endofunctors of sets. Its unit sends a point to its principal ultrafilter; its multiplication sends an ultrafilter of ultrafilters to the ultrafilter of subsets whose corresponding basic set of ultrafilters is large.

An adjunction $F⊣G$ with unit $\eta$ and counit $\epsilon$ induces the monadwith unit $\eta$.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

An algebra for a monad $(T,\eta ,\mu )$ is an object $A$ with a morphism $a:TA\to A$ satisfying $a{\eta }_A=1_A$ and $aT(a)=a{\mu }_A$.

Every algebra for a monad $(A,a)$ is the coequalizer, in the Eilenberg-Moore category, of $F(TA)\rightrightarrows F(A)$ with underlying arrows ${\mu }_A$ and $Ta$, followed by $a:F(A)\to (A,a)$. The pair has common section $F{\eta }_A$. If an algebra morphism $h:F(A)\to (B,b)$ equalizes the pair, the unique induced algebra morphism is $h{\eta }_A:(A,a)\to (B,b)$. This explicit proof does not require general colimits of algebras.

For an algebra for a monad $(X,a)$, its action $a:TX\to X$ satisfies $a{\eta }_X=1_X$.

For a monad $(T,\eta ,\mu )$, the free algebra functor $F:\mathcal{C}\to {\mathcal{C}}^T$ sends $X$ to $(TX,{\mu }_X)$ and $f$ to $Tf$. It is a left adjoint to the forgetful functor $U$, via $h\mapsto h{\eta }_X$ and $u\mapsto \gamma Tu$ between ${\mathcal{C}}^T(FX,(C,\gamma ))$ and $\mathcal{C}(X,C)$.

A morphism $f:(A,a)\to (B,b)$ of algebras for a monad satisfies $fa=bT(f)$. Such morphisms are the morphisms of the Eilenberg-Moore category.

The Eilenberg-Moore category ${\mathcal{C}}^T$ consists of algebras for a monad $T$ and their structure-preserving morphisms.

For algebra maps $f,g:(A,a)\rightrightarrows (B,b)$ whose underlying maps have a split coequalizer $q:B\to Q$, every functor preserves the split diagram, including $T$ and $T^2$. Since $qb$ equalizes $Tf,Tg$, it factors uniquely to the displayed action $c:TQ\to Q$. Cancellation of the epimorphisms $q,Tq,T^{2}q$ proves the unit and multiplication laws. Any algebra-valued map equalizing the pair factors through $q$, and cancelling $Tq$ proves that its factor preserves the action. This establishes creation of that coequalizer.

Given a diagram of algebras for a monad, take any chosen limit $L$ of its underlying diagram. The displayed equations uniquely determine $a:TL\to L$, because their right-hand sides form a cone. Joint monicity of the limit projections proves $a{\eta }_L=1$ and $aT(a)=a{\mu }_L$. The same projection argument proves that every induced mediating map preserves the algebra action. Thus the forgetful functor strictly creates this limit with the prescribed underlying cone, without assuming the monad preserves limits.

For a monad $(T,\eta ,\mu )$, the free algebra functor sends $X$ to $(TX,{\mu }_X)$ and is a left adjoint to the forgetful functor from the Eilenberg-Moore category. The transpose bijection sends an monad algebra morphism $h:TX\to A$ to $h{\eta }_X$; its inverse sends $k:X\to A$ to $aT(k)$, where $a:TA\to A$ is the algebra structure. The monad and algebra laws make these inverse and natural. The adjunction unit is $\eta$ and the adjunction counit at $(A,a)$ is $a$, so the induced monad has exactly the original endofunctor, unit and multiplication.

For a functor between Eilenberg-Moore categories lying over a right adjoint on the base categories and compatible with the monad structures, an adjoint lifting theorem constructs a left adjoint when the required reflexive coequalizers of algebra presentations exist. Applied through power-object monadicity, this converts a left adjoint of a logical functor into a right adjoint of that logical functor. The coequalizer hypotheses are part of the theorem; commutation with the forgetful functors alone is insufficient.

For algebras for a monad $(A,\alpha ),(B,\beta )$ and underlying binary coproducts in a category, put $\kappa =[T{\nu }_1,T{\nu }_2]:TA+TB\to T(A+B)$. If the algebra pair $F(\alpha +\beta ),{\mu }_{A+B}F\kappa :F(TA+TB)\rightrightarrows F(A+B)$ has a coequalizer, that coequalizer is their algebra coproduct. An algebra map from $F(A+B)$ transposes to $[a,b]:A+B\to C$; equalizing the pair is exactly the two algebra-morphism equations for $a,b$. The pair is reflexive through $F({\eta }_A+{\eta }_B)$.

If $\mathcal{C}$ has finite colimits and a monad $T$ preserves reflexive coequalizers, its Eilenberg-Moore category has finite colimits. The forgetful functor creates reflexive coequalizers preserved by $T$. The coproduct presentation for monad algebras yields binary coproducts, and $F0$ is initial. Any pair $a,b:X\rightrightarrows Y$ can then be replaced by the reflexive pair $[a,1],[b,1]:X+Y\rightrightarrows Y$, with identical coequalizers. Finite coproducts and coequalizers give all finite colimits.

For $F⊣G:\mathcal{D}\to \mathcal{C}$ with induced monad $T=GF$, the Eilenberg-Moore comparison functor is

If each standard counit presentation is a coequalizer, the Eilenberg-Moore comparison functor is full and faithful. An monad algebra morphism $\alpha :GB\to GC$ makes ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

For an adjunction $F⊣G$, applying $G$ to its standard counit presentation gives $T^{2}GD\rightrightarrows TGD\to GD$, split by the monad units. Reflection therefore makes every counit presentation a coequalizer and gives full faithfulness by descent. Conversely, monad algebra forgetful functor creates split coequalizers makes an existing G-split quotient a coequalizer in the Eilenberg-Moore category. Fullness and faithfulness of the comparison lift its universal factorization back to the original category.

The free-forgetful adjunction for the Eilenberg-Moore category is terminal in the category of adjunctions inducing a fixed monad. The unique morphism into it is the Eilenberg-Moore comparison functor $B\mapsto (GB,G{\epsilon }_B)$. Its underlying objects and arrows are forced by the forgetful functor, and counit compatibility forces its algebra actions.

If $F⊣G:\mathcal{D}\to \mathcal{C}$ induces $T=GF$ and $\mathcal{D}$ has coequalizers of reflexive pairs, then the comparison $K:\mathcal{D}\to {\mathcal{C}}^T$ has a left adjoint. On a $T$-algebra $(A,a)$ it is the coequalizerThe pair is reflexive through $F{\eta }_A$, and its universal property gives $L⊣K$.

For $F⊣G$ inducing $T=GF$, the Eilenberg-Moore comparison functor has a left adjoint exactly when every algebra $(A,a)$ has a coequalizer $q_a:FA\to L(A,a)$ of $Fa,{\epsilon }_{FA}:FTA\rightrightarrows FA$. Transposition identifies maps equalizing this pair with monad algebra morphisms $(A,a)\to K(D)$, proving both directions of the criterion. The unit is $\lambda =G{q}_a{\eta }_A$ and satisfies $G{q}_a=\lambda a$. In the base category, $a:TA\to A$ is a split coequalizer of $Ta,{\mu }_A:T^{2}A\rightrightarrows TA$, split by ${\eta }_A$ and ${\eta }_{TA}$. Consequently $\lambda$ is invertible exactly when $G{q}_a$ is also a coequalizer. The unit is invertible everywhere precisely when $G$ preserves all these specified coequalizers; preservation of arbitrary coequalizers is not required.

When each comparison functor has a left adjoint, iterating the Eilenberg-Moore comparison functor produces the monadic tower. Its monadic length is the least number of comparison steps needed to reach an equivalence; an equivalence has length zero and a non-equivalence that is already monadic has length one.

For the nested partial unary operation category tower, each one-step forgetful functor is a monadic adjunction right adjoint: it reflects isomorphisms and creates its split coequalizers. The Beck monadicity theorem identifies the first Eilenberg-Moore category with the next category in the tower. A long composite has that same first monad because free extension of nested partial unary operations leaves no new top fixed points. Repeating the comparison forgets one fewer operation at each step. The remaining forgetful functor is not full until no operation remains to forget, giving monadic length $m-n$.

An adjunction is monadic when its comparison functor from the right-hand category to the Eilenberg-Moore category of the induced monad is an equivalence.

The induced monad determines this comparison; Beck's monadicity theorem gives a criterion for it to be an equivalence of categories.

A useful form of the crude monadicity theorem says that a right adjoint is monadic if it reflects isomorphisms and its source has and the functor preserves coequalizers of reflexive pairs.

An adjunction is comonadic when the comparison from its left-hand category to coalgebras for the induced comonad is an equivalence. The dual Beck theorem tests this using reflected isomorphisms and suitable equalizers.

The relevant criterion is the Beck comonadicity theorem, dual to Beck's monadicity theorem.

The Beck comonadicity theorem, dual to the Beck monadicity theorem, says that a left adjoint is comonadic when it reflects isomorphisms and preserves the required equalizers of pairs whose images admit split equalizers. A stronger convenient hypothesis is that it preserves all equalizers.

The precise monadicity theorem says that a right adjoint is monadic exactly when it reflects isomorphisms and creates coequalizers of the pairs whose images have split coequalizers.

If an endofunctor $F$ has a right adjoint $G$ and $(F,\eta ,\mu )$ is a monad, the mate correspondence turns $\eta :1\to F$ and $\mu :F^2\to F$ into a counit $G\to 1$ and comultiplication $G\to G^2$. The reversed mate correspondence turns the monad laws into the comonad laws and identifies $F$-algebras with $G$-coalgebras.

For a monad $(T,\eta ,\mu )$ on $\mathcal{C}$, the Kleisli category ${\mathcal{C}}_T$ has the objects of $\mathcal{C}$ and morphismsIts identity on $A$ is ${\eta }_A$, and the composite of $f:A\to TB$ and $g:B\to TC$ is ${\mu }_{C}T(g)f:A\to TC$.

For a monad $(T,\eta ,\mu )$, the free functor $J:\mathcal{C}\to {\mathcal{C}}_T$ and functor $U:{\mathcal{C}}_T\to \mathcal{C}$ with $UA=TA$ and $U(f)={\mu }_{B}T(f)$ form an adjunction inducing that monad. Every other adjunction inducing the same monad receives the unique Kleisli comparison functor commuting with its left and right adjoints and preserving the adjunction structure. This is initiality among adjunctions inducing the specified monad, with morphisms required to respect that structure.

If $L⊣R$ induces a monad $T=RL$, the comparison maps $A$ to $LA$ and maps a Kleisli category arrow $f:A\to TB$ to ${\epsilon }_{LB}L(f)$. It is full and faithful, by the adjunction bijection $\mathcal{D}(LA,LB)\cong \mathcal{C}(A,TB)$. Consequently it is part of an equivalence of categories precisely when every object of $\mathcal{D}$ is isomorphic to some $LA$.

The free functor $F_T:\mathcal{C}\to {\mathcal{C}}_T$ is the identity on objects and sends $f:A\to B$ to ${\eta }_{B}f:A\to TB$. A functor $G:\mathcal{C}\to \mathcal{D}$ carries an algebra for the precomposition monad $G\mapsto GT$ exactly when it factors through $F_T$.

For a monad $T$ on $\mathcal{C}$ and a category $\mathcal{D}$, postcomposition defines a monad $T_{\ast }:[\mathcal{D},\mathcal{C}]\to [\mathcal{D},\mathcal{C}]$ whose unit and multiplication are pointwise. Its Eilenberg-Moore category is canonically equivalent to $[\mathcal{D},{\mathcal{C}}^T]$.

Precomposition defines a monad $T^{\ast }:[\mathcal{C},\mathcal{D}]\to [\mathcal{C},\mathcal{D}]$ by $G\mapsto GT$, with unit $G\eta$ and multiplication $G\mu$. Its Eilenberg-Moore category is canonically equivalent to $[{\mathcal{C}}_T,\mathcal{D}]$.