Alphabeta MathDefinitionDefinition: 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.
Kleisli category of a monad
Definition
For a monad on , the Kleisli category has the same objects as and hom-collections
The identity at is , and composition is . The category laws are established before this definition in Kleisli composition is associative and unital.
Depends on
Used by
- A Kleisli composite for the list monad computed by substitution and concatenation Example
- The Kleisli category of the maybe monad is the category of sets and partial functions Example
- FALSE: The Kleisli and Eilenberg–Moore categories are equivalent for every monad False statement
- The Kleisli adjunction induces the given monad Theorem
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 5.2.10 (standard reference, not scraped)
- B. Richter, From Categories to Homotopy Theory, Definition 6.3.1 (standard reference, not scraped)