DefinitionDefinition: 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 ordinary category of an enriched category

Definition

Let $A$ be a $V$ -category (Enriched category over a monoidal base).

Its underlying ordinary category $A_{0}$ has the same objects as $A$ and hom-sets

$A_{0} (A, B) : = V (1, A (A, B)) .$

So an ordinary morphism $A \to B$ in $A_{0}$ is a global element of the hom-object $A (A, B)$ .

Composition in $A_{0}$ is induced from enriched composition: if $f : 1 \to A (A, B)$ and $g : 1 \to A (B, C)$ , then

$g \circ_{0} f : = 1 \overset{≅}{\to} 1 \otimes 1 \overset{g \otimes f}{\to} A (B, C) \otimes A (A, B) \overset{M_{A, B, C}}{\to} A (A, C) .$

The identity morphism of $A$ in $A_{0}$ is the enriched identity $j_{A} : 1 \to A (A, A)$ .

Depends on

Enriched category over a monoidal base

Used by

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, equations (1.10) and (1.11) (standard reference, not scraped)
Emily Riehl, Categorical Homotopy Theory, Section 3.4 (standard reference, not scraped)