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

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 $$g_\Kl \compL_\Kl f_\Kl := \mu \compL Tg_\Kl \compL f_\Kl$$

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

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

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

## 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

• $$LA$$ 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

• $$[[A, R], R]$$

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 $$\cC$$