Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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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: every monoidal category is strict

Statement

False claim: every monoidal category is strict.

Facts & Assumptions

Given: A skeleton of Set containing an infinite set D with D≅D×D.

[L1]

A strict monoidal category makes associativity and unit equalities literal and all constraints identities (Strict monoidal category).

[L2]

In the Isbell skeleton example, forcing those identities collapses every endomorphism D→D to the same map, which is absurd (Isbell's warning that isomorphic objects cannot simply be identified).

Refutation

technique · direct
1.1

If every monoidal category were strict, then the monoidal structure on that skeleton of Set would satisfy the condition in [L1].

L1
1.2

But [L2] says that in this example such an identification forces all endomorphisms D→D to be equal, contradicting the existence of distinct endomorphisms such as the identity and a constant map.

L2
2.1

Therefore not every monoidal category is strict.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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