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

Statement

False claim: for every monad, its Kleisli and Eilenberg–Moore categories are equivalent.

The free-monoid monad on $S e t$ is a counterexample.

Facts & Assumptions

Given: The free-monoid monad $T (X) = X^{*}$ on $S e t$ .

[L1]

Its Kleisli hom-set from $X$ to $Y$ is $S e t (X, Y^{*})$ (Kleisli category of a monad; The free-monoid monad has monoids as its Eilenberg–Moore algebras).

[L2]

Its Eilenberg–Moore category is the category of monoids (The free-monoid monad has monoids as its Eilenberg–Moore algebras).

[L3]

A functor $F : C \to D$ 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); an equivalence consists of quasi-inverse functors and natural isomorphisms $1_{C} \Rightarrow G F$ and $F G \Rightarrow 1_{D}$ (Equivalence, quasi-inverse, and adjoint equivalence of categories), and $F$ is fully faithful when every $F_{A, B} : C (A, B) \to D (F A, F B)$ is bijective (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[L4]

The canonical Kleisli comparison is fully faithful with image the free algebras (The comparison from the Kleisli category is fully faithful with image the free algebras).

[L5]

For an idempotent monad $T$ , the canonical comparison $M : C_{T} \to C^{T}$ is an equivalence of categories (The Kleisli and Eilenberg–Moore categories of an idempotent monad are equivalent).

Refutation

technique · direct

1.1L1

By [L1], the endomorphism set of $X$ in the Kleisli category is $S e t (X, X^{*})$ .

1.2L2

The monoid $M = {1, e}$ with $e^{2} = e$ has exactly two endomorphisms: an endomorphism fixes $1$ , and it may send $e$ to either idempotent $1$ or $e$ .

2.1step 1.1

If $X = \emptyset$ , this function set is a singleton. If $X$ is nonempty, choose $x \in X$ ; the words $[], [x], [x, x], \dots$ are distinct, and the corresponding constant functions show that $S e t (X, X^{*})$ is infinite.

3.1L2L3step 1.2step 2.1

If the two categories were equivalent, essential surjectivity would place $M$ in the image up to isomorphism, and full faithfulness would give a bijection between its two-element endomorphism set and the endomorphism set of some Kleisli object. Step 2.1 rules this out.

4.1L4L5step 3.1∎

The claim is therefore false. The positive boundary is [L4], that the canonical comparison is fully faithful with image the free algebras, together with [L5], that it is an equivalence when the monad is idempotent.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 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 5.2.14 (standard reference, not scraped)