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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

1.1given

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 $U_{T} (g) = μ_{B} \circ T (g) : T A \to T B$ .

2.1step 1.1given

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.

3.1step 1.1step 2.1∎

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 $μ$ .

Depends on

Used by

Dependency tree · two levels

5 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 5.2.12 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Corollary 6.3.6 and Remark 6.3.7 (standard reference, not scraped)