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.

Monadic and strictly monadic functors

Definition

Let $U : D \to C$ have a left adjoint $F$ , let $T = U F$ be the induced monad, and let $K : D \to C^{T}$ be the comparison functor of The comparison functor to the Eilenberg–Moore category exists and is unique. The functor $U$ is monadic when $K$ is an equivalence of categories (Equivalence, quasi-inverse, and adjoint equivalence of categories).

It is strictly monadic when $K$ is an isomorphism of categories, so its object and morphism correspondences, inverse, and equations hold on the nose. Strict monadicity implies monadicity; the converse is not part of the definition.

Depends on

Used by

Every category monadic over Set is complete Corollary
Every monadic functor is conservative Theorem
The inclusion of a reflective full subcategory is monadic Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 12 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 5.3.1 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Chapter VI (standard reference, not scraped)