TheoremStatement: AI-adaptedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Kleisli composition is associative and unital

Statement

Let $(T, η, μ)$ be a monad on $C$ . For morphisms $f : A \to T B$ and $g : B \to T C$ , define

$g ⋆ f : = μ_{C} \circ T (g) \circ f : A \to T C .$

Then $⋆$ is associative, and $η_{A} : A \to T A$ is a two-sided identity at $A$ .

Facts & Assumptions

Given: A monad $(T, η, μ)$ as in Monad on a category and composable arrows $f : A \to T B$ , $g : B \to T C$ , and $h : C \to T D$ .

Proof

technique · direct

1.1

Expanding the definitions gives $(h ⋆ g) ⋆ f = μ_{D} \circ T (μ_{D}) \circ T^{2} (h) \circ T (g) \circ f$ , while $h ⋆ (g ⋆ f) = μ_{D} \circ T (h) \circ μ_{C} \circ T (g) \circ f$ .

given

2.1

Naturality of $μ$ at $h$ rewrites $T (h) \circ μ_{C}$ as $μ_{T D} \circ T^{2} (h)$ , and the monad associativity equation $μ_{D} \circ T (μ_{D}) = μ_{D} \circ μ_{T D}$ identifies the two expansions in step 1.1.

step 1.1given

3.1

Naturality of $η$ and the two distinct unit laws give $f ⋆ η_{A} = μ_{B} \circ T (f) \circ η_{A} = μ_{B} \circ η_{T B} \circ f = f$ and $η_{B} ⋆ f = μ_{B} \circ T (η_{B}) \circ f = f$ , so the asserted identities hold.

given∎

Depends on

Monad on a category

Used by

Kleisli category of a monad Definition
Co-Kleisli composition is associative and unital Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, 2nd ed., Example 5.2.iv (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Example 6.3.2 (standard reference, not scraped)