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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

1.1given

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

1.2given

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 $a$ .

2.1step 1.1step 1.2L1∎

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.

Depends on

Used by

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

Dependency tree · two levels

18 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 5.3.i (standard reference, not scraped)