TheoremStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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The Kleisli adjunction induces the given monad

Statement

For a monad $(T, η, μ)$ on $C$ , there is an adjunction

$F_{T} : C ⇄ C_{T} : U_{T}$

whose induced monad is $(T, η, μ)$ on the nose.

Facts & Assumptions

Given: The Kleisli category $C_{T}$ of Kleisli category of a monad and the monad $(T, η, μ)$ used to define it.

Proof

technique · direct

1.1

Define $F_{T} A = A$ and $F_{T} (f) = η_{B} \circ f : A \to T B$ . Define $U_{T} A = T A$ , and for a Kleisli arrow $g : A \to T B$ put $U_{T} (g) = μ_{B} \circ T (g) : T A \to T B$ .

given

2.1

The Kleisli identity and associativity laws show that both assignments preserve identities and composition. Moreover $C_{T} (F_{T} A, B) = C (A, T B) = C (A, U_{T} B)$ naturally in $A$ and $B$ , so they form an adjunction.

step 1.1given

3.1

The unit is $η$ . The counit at a Kleisli object $B$ is represented by $1_{T B} : T B \to T B$ , and applying $U_{T}$ gives $μ_{B}$ ; the Kleisli unit laws are the triangle identities. Thus $U_{T} F_{T} = T$ and the induced unit and multiplication are $η$ and $μ$ .

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 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., Lemma 5.2.12 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Corollary 6.3.6 and Remark 6.3.7 (standard reference, not scraped)