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.

Morphisms between monads on one category

Definition

Let $(T, η^{T}, μ^{T})$ and $(S, η^{S}, μ^{S})$ be monads on the same category $C$ . A monad morphism $α : T \Rightarrow S$ is a natural transformation satisfying

$α \circ η^{T} = η^{S}$

and

$α \circ μ^{T} = μ^{S} \circ S α \circ α T .$

At an object $A$ , the multiplication equation is

$α_{A} μ_{A}^{T} = μ_{A}^{S} S (α_{A}) α_{T A} .$

Depends on

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

M. Richter, From Categories to Homotopy Theory, Definition 6.1.15 (standard reference, not scraped)