TheoremStatement: AI-adaptedProof: Literature-sourcedprecheck 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

1.1given

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

2.1step 1.1given

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.

3.1given∎

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.

Depends on

Monad on a category

Used by

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

Dependency tree · two levels

4 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., Example 5.2.iv (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Example 6.3.2 (standard reference, not scraped)