ExampleConstruction: 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Strictification of a cartesian monoidal category computed

Example

Let $C$ be a category with finite products, so that $C$ is monoidal under $\times$ by A category with finite products is monoidal. Its strictification sends an object $X$ to the endofunctor $X \times -$ .

Facts & Assumptions

Given: A category $C$ with finite products.

[L1]

Finite products make $C$ into a monoidal category with tensor $\times$ (A category with finite products is monoidal).

[L2]

Strictification sends $X$ to the right-module endofunctor $X \otimes -$ and is monoidally equivalent to a strict monoidal category (Mac Lane strictification).

Verification

technique · direct

1.1

By [L1], the tensor product is $\times$ , so the strictification formula from [L2] becomes $L (X) (Y) = X \times Y$ . Its right-module structure map at $(Y, Z)$ is the cartesian associator $(X \times Y) \times Z \to X \times (Y \times Z)$ .

L1L2

2.1

The binary comparison $L (X) \otimes L (Y) \Rightarrow L (X \times Y)$ therefore has component the inverse reassociation $X \times (Y \times Z) \to (X \times Y) \times Z$ , and the unit comparison is $Z \to 1 \times Z$ .

step 1.1L2

3.1

So in the cartesian case the abstract strictification is completely explicit: it packages ordinary reassociation maps of products into a strict monoidal category of endofunctors.

step 2.1∎

Depends on

Used by

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Dependency tree · two levels

13 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

S. Mac Lane, Categories for the Working Mathematician, Chapter XI.3 (standard reference, not scraped)
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Chapter 2.8 (standard reference, not scraped)