False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: every lax monoidal functor has invertible structure maps

Statement

False claim: every lax monoidal functor has invertible structure maps.

Facts & Assumptions

Given: The page's distinction between lax and strong monoidal functors and the cartesian monoidal category $S e t$ .

[L1]

A strong monoidal functor is a lax monoidal functor whose binary and unit structure maps are isomorphisms (Lax, strong, and strict monoidal functors).

[L2]

Different sources use the bare phrase "monoidal functor" differently, so this library avoids the bare phrase precisely to prevent that conflation (Why the bare phrase 'monoidal functor' is ambiguous across sources).

[L3]

The power set $P (X)$ consists of all subsets of $X$ (The power set $P (x) = {z : z \subseteq x}$ ).

[L4]

The category $S e t$ is cartesian monoidal (Set, Cat, and every complete category are cartesian monoidal).

Refutation

technique · direct

1.1

Let $P : S e t \to S e t$ send a function to its direct-image map on power sets. Define $P_{2} (A, B) : = A \times B$ and let $P_{0} : 1 \to P (1)$ select the full singleton. Direct images preserve identities and composition, and $(f \times g) (A \times B) = f (A) \times g (B)$ proves naturality. Both associativity composites send $(A, B, C)$ to $A \times B \times C$ with the indicated bracketing, and the unit composites return $A$ . Thus $P$ is lax monoidal.

L3L4constructalgebra

2.1

For $X = Y = {0, 1}$ , the diagonal ${(0, 0), (1, 1)} \subseteq X \times Y$ is not $A \times B$ for any $A \subseteq X$ and $B \subseteq Y$ . Hence $P_{2}$ is not surjective, so $P$ is not strong by [L1].

L1step 1.1algebra

3.1

The distinction in [L1] is therefore genuine, and the stated universal claim is false. The qualifier "lax" makes the claim source-stable, while [L2] explains why the bare phrase "monoidal functor" would not.

L1L2step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Definition 2.4.1 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, Chapter XI.2 (standard reference, not scraped)