CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

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A skeleton of Set cannot be made strict by identifying isomorphic objects

Counterexample

Let $S$ be a skeleton of $S e t$ and choose an infinite object $D \in S$ with $D ≅ D \times D$ . Then $S$ witnesses that one cannot obtain a strict monoidal structure merely by identifying isomorphic objects.

Facts & Assumptions

Given: A skeleton $S$ of $S e t$ containing such an object $D$ .

[L1]

In Isbell's skeleton argument, forcing the associator to become an identity collapses every endomorphism $D \to D$ to the same map, which is absurd (Isbell's warning that isomorphic objects cannot simply be identified).

Verification

technique · counterexample

1.1

Assume one could make the monoidal structure on $S$ strict merely by identifying isomorphic objects. Then the associator at $D$ would be an identity in the sense excluded by [L1].

givenL1

1.2

By [L1], that would force all endomorphisms $D \to D$ to be equal. But $1_{D}$ and any constant endomorphism are different maps, since $D$ has at least two elements.

2.1

Hence $S$ is a counterexample to the claim.

step 1.1step 1.2∎

Depends on

Isbell's warning that isomorphic objects cannot simply be identified

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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.