DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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.

Subcategory and full subcategory

Definition

A subcategory $\mathcal A$ of a category $\mathcal C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) has a subclass of the objects of $\mathcal C$ and, for each $A,B\in\mathcal A$ , a subclass $\mathcal A(A,B)\subseteq\mathcal C(A,B)$ . It contains the identities of its objects and is closed under the composition inherited from $\mathcal C$ .

The subcategory is full when $\mathcal A(A,B)=\mathcal C(A,B)$ for every pair of its objects. Thus a full subcategory is determined entirely by its objects.

Depends on

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)