TheoremStatement: AI-adaptedProof: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1given

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

1.2L1given

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 $2 \Rightarrow 3$ .

2.1given∎

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 $3 \Rightarrow 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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 5.1.iii (standard reference, not scraped)