DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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.

Dense subcategory

Definition

Let $i : A \to C$ be a fully faithful functor with $A$ small and $C$ locally small (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors, Small, locally small, and large categories). The functor $i$ is dense when the identity functor $1_{C} : C \to C$ is a pointwise left Kan extension of $i$ along $i$ in the sense of Pointwise Kan extensions by the comma-category formula.

Equivalently, for each object $c$ of $C$ , the canonical diagram indexed by the category of elements of the presheaf $C (i -, c)$ (The category of elements of a covariant functor or a presheaf) has colimit $c$ . The model case is the Yoneda embedding: it is fully faithful by The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective and satisfies the pointwise self-extension property by The Yoneda embedding is its own pointwise left Kan extension.

When $A$ is identified with a full subcategory of $C$ and $i$ is the inclusion, one also says that $A$ is a dense subcategory of $C$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

B. Richter, From Categories to Homotopy Theory, Definition 5.4.1 (standard reference, not scraped)
E. Riehl, Category Theory in Context, 2nd ed., §6.5 (standard reference, not scraped)