Questions: Monads in Category Theory

Question 1

In the Kleisli category C_T for a monad (T, η, μ), a morphism from A to B is a C-morphism f: A → TB. How is the Kleisli composition of f: A → TB and g: B → TC defined?

• A. Apply g to f's output directly: g ∘ f : A → TC, since g takes B-values
• B. Apply T to g to get Tg: TB → T(TC), compose with f to get Tg ∘ f: A → T(TC), then apply μ_C: T(TC) → TC
• C. Apply μ first to collapse T² and then compose f with g in the original category
• D. Kleisli composition is just ordinary composition in C with no modification needed

Answer: (B)

Kleisli composition requires lifting g through the functor T and then flattening with μ. Given f: A → TB and g: B → TC, you cannot directly compose them (the codomain of f is TB, not B). Instead: apply T to g to get Tg: TB → T(TC), compose with f to get Tg ∘ f: A → T(TC), then apply μ_C: T(TC) → TC to flatten. The result is μ_C ∘ Tg ∘ f: A → TC. In Haskell, this is precisely the bind (>>=) operation.

Question 2

A monad on a category C consists of which of the following structures?

• A. An endofunctor T: C → C alone, which automatically carries monad structure
• B. Two functors F: C → D and G: D → C forming an adjunction
• C. An endofunctor T: C → C together with natural transformations η: Id_C ⇒ T (unit) and μ: T∘T ⇒ T (multiplication) satisfying associativity and unit laws
• D. A functor T: C → C and a single natural transformation μ: T∘T ⇒ T

Answer: (C)

A monad is the triple (T, η, μ): an endofunctor, a unit embedding objects into the T-context, and a multiplication flattening T² to T. Both η and μ are required, and the monad laws must be verified — they are not automatic. An endofunctor alone carries no monad structure. An adjunction gives rise to a monad but is more data than a monad itself. Option D omits the unit, which is essential.

Question 3

Every adjunction F ⊣ G: C ⇄ D gives rise to a monad on C via T = G ∘ F.

• T. True
• F. False

Answer: True

This is a fundamental theorem. Given F ⊣ G, the composite T = G ∘ F: C → C is an endofunctor. The adjunction unit η: Id_C ⇒ GF serves as the monad unit. The counit ε: FG ⇒ Id_D gives the monad multiplication μ = GεF: GFGF ⇒ GF (i.e., μ: T² ⇒ T). The monad laws follow from the triangle identities of the adjunction. Every free-forgetful adjunction (groups, rings, etc.) produces a monad this way.

Question 4

Any endofunctor T: C → C equipped with a natural transformation η: Id_C ⇒ T automatically constitutes a monad.

• T. True
• F. False

Answer: False

A monad requires both a unit η: Id_C ⇒ T and a multiplication μ: T² ⇒ T satisfying the associativity and unit laws. An endofunctor with only a unit is far from sufficient — there may be no natural transformation μ: T² ⇒ T at all, or if one exists, it may fail the monad laws. The laws are genuinely nontrivial coherence conditions that must be verified. Not every endofunctor is a monad.

Question 5

Explain why a monad is called 'a monoid in the category of endofunctors' and identify which monad structures play the roles of unit element and multiplication.

Model answer: A monoid is a set with an associative binary operation and an identity element. In the category of endofunctors [C, C] (with functor composition as the monoidal product), a monad (T, η, μ) makes T a monoid object: η: Id ⇒ T is the 'unit element' (the identity functor Id plays the role of the identity element), and μ: T∘T ⇒ T is the 'multiplication' (composing two copies of T into one). The monad laws (μ ∘ Tμ = μ ∘ μT; μ ∘ Tη = id = μ ∘ ηT) are exactly the monoid associativity and unit laws stated as equations between natural transformations.

This description is genuinely precise, not just a slogan. The monoidal category is [C, C] with composition (∘) as tensor product and Id_C as the unit object. A monoid object in any monoidal category (M, ⊗, I) is an object m with morphisms μ: m ⊗ m → m and η: I → m satisfying associativity and unit laws. Setting m = T, ⊗ = ∘, and I = Id_C gives the monad definition exactly. Understanding monads as monoid objects in [C, C] explains why monads appear wherever algebras do and lets category theorists generalize the construction to other monoidal settings.