TheoremStatement: AI-adaptedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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For a monad, invertibility of multiplication, monicity of every multiplication component, and equality of the two whiskered units are equivalent

Statement

For a monad $(T, η, μ)$ , the following are equivalent:

$μ$ is a natural isomorphism, so the monad is idempotent (Idempotent monad);
every component $μ_{A} : T^{2} A \to T A$ is monic;
$η T = T η : T \Rightarrow T^{2}$ .

Facts & Assumptions

Given: A monad $(T, η, μ)$ .

[L1]

A monomorphism is left-cancellable: $m \circ g = m \circ h$ implies $g = h$ (Monomorphism and epimorphism by left and right cancellation).

Proof

technique · direct

1.1

If $μ$ is a natural isomorphism, each $μ_{A}$ is an isomorphism and therefore monic, proving $1 \Rightarrow 2$ .

given

1.2

If every $μ_{A}$ is monic, the monad unit laws give $μ_{A} \circ η_{T A} = 1_{T A} = μ_{A} \circ T (η_{A})$ ; cancellation by [L1] yields $η_{T A} = T (η_{A})$ for every $A$ , proving $2 \Rightarrow 3$ .

L1given

2.1

Assume $η T = T η$ . The unit law makes $η_{T A}$ a right inverse to $μ_{A}$ . Naturality of $η$ at $μ_{A}$ , followed by $η_{T^{2} A} = T (η_{T A})$ , gives $η_{T A} \circ μ_{A} = T (μ_{A}) \circ T (η_{T A}) = 1_{T^{2} A}$ ; thus $η_{T A}$ is also a left inverse, every $μ_{A}$ is invertible, and $3 \Rightarrow 1$ .

given∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 7 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 5.1.iii (standard reference, not scraped)