False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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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.

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FALSE: A monad is a monoid object in the endofunctor category for every category

Statement

False claim under the library's size convention: for every category $C$ , a monad on $C$ is a monoid object in the endofunctor category $[C, C]$ .

The slogan is valid when that endofunctor category is formed, as recorded in The monoid description of a monad requires an endofunctor category.

Facts & Assumptions

Given: The library convention for functor categories.

[L1]

The functor category $[C, D]$ is formed when the source $C$ is small; for an arbitrary large source the same notation may be used only as metatheoretic shorthand, and the definition does not form those proper-class-sized data into a category (Functor category $[C, D]$ ).

[L2]

If $C$ is small and $D$ is locally small, then $[C, D]$ is locally small; if both are small, then $[C, D]$ is small (If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L3]

Sets as objects and functions as morphisms form a large locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[L4]

A monad on $C$ is an endofunctor with a unit and a multiplication satisfying the two unit equations and associativity (Monad on a category).

Refutation

technique · direct

1.1

A monoid object is defined only inside an actual monoidal category, so the claimed description presupposes that $[C, C]$ is a category.

2.1

Take $C = S e t$ , which is large by [L3], carrying the identity monad $(1_{S e t}, 1, 1)$ , whose unit and associativity equations hold trivially by [L4]. The source $S e t$ is not small, so by [L1] the adopted convention does not form $[S e t, S e t]$ into a category, and the presupposition of step 1.1 fails for this monad.

L1L3L4step 1.1

3.1

A claim asserted for every category therefore fails at $C = S e t$ , where it presupposes a category the convention does not form. When $C$ is small the functor category is formed by [L1] and is locally small by [L2], and the usual monoid description is valid there.

L1L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 14 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., Remark 5.1.2 (standard reference, not scraped)