RemarkRemark: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The underlying category can lose genuinely enriched information

Remark

Passing from $A$ to $A_{0}$ (The underlying ordinary category of an enriched category) keeps only the global elements of each hom-object. That can forget real structure.

For $V = C a t$ , the hom-object $A (A, B)$ is a whole category, but $A_{0} (A, B) = C a t (1, A (A, B))$ sees only its objects, so every 2-cell disappears. For differential graded or chain-complex enrichments, the same construction keeps only degree-zero cycles. Accordingly, an enriched limit, adjunction, or density statement may be strictly stronger than the corresponding statement in the underlying ordinary category.

Depends on

The underlying ordinary category of an enriched category

Used by

A bijection on underlying hom-sets need not exhibit a cotensor Counterexample
Enriched adjunction Definition
The underlying category of a Cat-enriched category forgets the 2-cells Example
FALSE: the underlying ordinary category determines the enriched category False statement
A conical enriched limit is stronger than a limit in the underlying category Theorem

Dependency tree · two levels

2 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

G. M. Kelly, Basic Concepts of Enriched Category Theory, Section 1.3 (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 3.4 (standard reference, not scraped)