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The Kleisli factorisation functor for an adjunction inducing a monad exists and is unique

Statement

Let $F : C ⇄ D : U$ be an adjunction with counit $ε$ whose induced monad is a fixed monad $T$ on $C$ on the nose, and write $F_{T} ⊣ U_{T}$ for the Kleisli adjunction with counit $ε^{T}$ . There is exactly one functor $J : C_{T} \to D$ satisfying

$J F_{T} = F, U J = U_{T}, J (ε_{B}^{T}) = ε_{F B} for every object B,$

namely

$J (A) = F (A), J (f : A \to T B) = ε_{F B} \circ F (f) : F A \to F B .$

These three equalities are what it means for $J$ to be a morphism of adjunctions from the Kleisli adjunction to $F ⊣ U$ .

Facts & Assumptions

Given: An adjunction $F ⊣ U$ as in Adjunction by unit, counit, and the triangle identities, inducing $T = U F$ with $μ = U ε F$ as in Every adjunction induces a monad on the domain of its left adjoint, and the Kleisli adjunction of The Kleisli adjunction induces the given monad.

Proof

technique · direct

1.1

A Kleisli arrow $f : A \to T B = U F B$ has adjoint transpose $ε_{F B} \circ F (f) : F A \to F B$ ; define $J$ by this formula and by $J (A) = F A$ .

given

2.1

For an identity $η_{A}$ , the first triangle identity gives $J (η_{A}) = ε_{F A} \circ F (η_{A}) = 1_{F A}$ . For $f : A \to T B$ and $g : B \to T C$ , substitution of $μ = U ε F$ , naturality of $ε$ , and the triangle identities gives $J (g ⋆ f) = J (g) \circ J (f)$ , so $J$ is a functor.

step 1.1given

3.1

The formulas give $J F_{T} = F$ and $U J = U_{T}$ on objects and arrows, and $J$ sends the Kleisli counit to $ε$ . Conversely, those equalities force the image of $f : A \to U F B$ to be its adjoint transpose, so they force the formula of step 1.1 on every arrow and prove uniqueness.

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 10 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., Proposition 5.2.13 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Theorem 6.3.10 (standard reference, not scraped)