TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Every adjunction induces a monad on the domain of its left adjoint

Statement

Let $F : C \to D$ be left adjoint to $G : D \to C$ , with unit $η : 1_{C} \Rightarrow G F$ and counit $ε : F G \Rightarrow 1_{D}$ . Then

$T : = G F, η : 1_{C} \Rightarrow T, μ : = G ε F : T^{2} \Rightarrow T$

define a monad on $C$ (Monad on a category).

Facts & Assumptions

Given: An adjunction $F ⊣ G$ with unit $η$ and counit $ε$ as in Adjunction by unit, counit, and the triangle identities.

[L1]

The triangle identities are $(ε F) \circ (F η) = 1_{F}$ and $(G ε) \circ (η G) = 1_{G}$ (Adjunction by unit, counit, and the triangle identities).

Proof

technique · direct

1.1given

Put $T = G F$ and $μ = G ε F$ . Whiskering preserves naturality, so $T$ is an endofunctor and $η : 1_{C} \Rightarrow T$ and $μ : T^{2} \Rightarrow T$ are natural transformations.

2.1givenstep 1.1L1

At an object $A$ , the two associativity composites are $μ_{A} \circ T (μ_{A}) = G (ε_{F A} \circ F G (ε_{F A}))$ and $μ_{A} \circ μ_{T A} = G (ε_{F A} \circ ε_{F G F A})$ ; naturality of $ε$ at $ε_{F A} : F G F A \to F A$ identifies the expressions inside $G$ , so $μ \circ T μ = μ \circ μ T$ .

3.1step 1.1L1∎

Componentwise, $μ_{A} \circ η_{T A} = G (ε_{F A}) \circ η_{G F A} = 1_{G F A}$ by the second triangle identity, while $μ_{A} \circ T (η_{A}) = G (ε_{F A} \circ F (η_{A})) = 1_{G F A}$ by the first; hence both unit laws hold and $(T, η, μ)$ is a monad.

Depends on

Used by

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Lemma 5.1.3 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Theorem 6.1.3 (standard reference, not scraped)