DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Free algebra for a monad

Definition

For a monad $(T, η, μ)$ and an object $A$ , the free $T$ -algebra on $A$ is

$(T A, μ_{A}) .$

It is a $T$ -algebra: its associativity axiom is the componentwise monad associativity equation, and its unit axiom is $μ_{A} \circ η_{T A} = 1_{T A}$ . For a morphism $f : A \to B$ , naturality of $μ$ makes $T (f) : (T A, μ_{A}) \to (T B, μ_{B})$ an algebra homomorphism.

Depends on

Algebra and algebra homomorphism for a monad

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 5 results over 5 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., Definition 5.2.8 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Example 6.2.3 (standard reference, not scraped)