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

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: $$\mu (T \eta) = \mu (\eta T) = \id_T$$

# 2 T-algebra

For a monad $$(T, \mu, \eta)$$, an algebra over a monad (T-algebra) is a left M-module $$(A, \alpha_A: TA \to A)$$ over the monad in the category of endofunctors.

Thus, a T-algebra is compatible with

• monad composition: $$\alpha_A \circ T \alpha_A = \alpha_A \circ \mu_A$$
• monad unit: $$\alpha_A \circ \eta_A = \id_A$$

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

# 3 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 $$g_\Kl \circ_\Kl f_\Kl := \mu \circ Tg_\Kl \circ f_\Kl$$

# 4 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.

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

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

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

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

# 7 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 $$\cC$$