Alphabeta MathHow 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.
FALSE: a monoid object in an endofunctor category is the definition of a monad
Statement
False claim: a monoid object in an endofunctor category is the definition of a monad.
Facts & Assumptions
Given: The endofunctor-category comparison theorem.
A monad is defined directly by an endofunctor and unit and multiplication natural transformations, without assuming that an endofunctor category exists (Monad on a category).
The equivalence with monoid objects in an endofunctor category is proved only when that category exists; this library forms it for small source categories (A monoid object in a small endofunctor category is exactly a monad, The monoid description of a monad requires an endofunctor category).
Sets and functions form the large category (Sets and functions form the large locally small category ).
Refutation
The identity endofunctor on the large category , with identity unit and multiplication, is a monad by [L1].
By [L3], is large, and [L2] says this library does not form its endofunctor category . Thus the monad in step 1.1 exists here although the proposed monoid-object formulation is unavailable.
Therefore the claim is false.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, Remark 5.1.2 (standard reference, not scraped)