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.

Algebra and algebra homomorphism for a monad

Definition

Let $(T, η, μ)$ be a monad on $C$ (Monad on a category). A $T$ -algebra is an object $A$ together with a morphism $a : T A \to A$ , called its structure map, satisfying

$a \circ η_{A} = 1_{A}, a \circ T (a) = a \circ μ_{A} .$

For $T$ -algebras $(A, a)$ and $(B, b)$ , a $T$ -algebra homomorphism $f : (A, a) \to (B, b)$ is a morphism $f : A \to B$ in $C$ such that

$f \circ a = b \circ T (f) .$

Depends on

Monad on a category

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 6 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 5.2.4 (standard reference, not scraped)
B. Richter, From Categories to Homotopy Theory, Definition 6.2.1 (standard reference, not scraped)