DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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.

Generalized elements and their shapes

Definition

Let $A$ and $S$ be objects of a category $C$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). A generalized element of $A$ of shape $S$ is a morphism $x : S \to A$ . The domain $S$ is its shape.

For a set $A$ , ordinary elements are exactly generalized elements of singleton shape: a function $1 \to A$ is determined by the image of the unique point, and every element of $A$ determines such a function. Other shapes can record more structure than singleton-shaped elements do.

Depends on

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

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.

Sources

Tom Leinster, Basic Category Theory, Definition 4.1.25 (standard reference, not scraped)