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The Kleisli and Eilenberg–Moore categories of an idempotent monad are equivalent

Statement

For an idempotent monad $T$ , the canonical comparison $M : C_{T} \to C^{T}$ is an equivalence of categories.

Facts & Assumptions

Given: The comparison theorem and the characterisation of algebras for an idempotent monad (The comparison from the Kleisli category is fully faithful with image the free algebras, Algebras for an idempotent monad form a reflective subcategory).

[L1]

A functor is an equivalence exactly when it is fully faithful and split essentially surjective (A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice).

Proof

technique · direct

1.1

The canonical comparison is fully faithful, and its strict image consists of the free algebras.

given

1.2

For each algebra $(A, a)$ , the idempotent-algebra theorem gives $a = η_{A}^{- 1}$ . The map $η_{A} : (A, a) \to (T A, μ_{A})$ is therefore a specified algebra isomorphism from $(A, a)$ to the free algebra on $A$ , with inverse $a$ .

given

2.1

The specified isomorphisms of step 1.2 make the fully faithful comparison of step 1.1 split essentially surjective; [L1] therefore makes it an equivalence.

step 1.1step 1.2L1∎

Depends on

Used by

FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 14 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 5.3.i (standard reference, not scraped)