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.

Final and initial functors via nonempty connected comma categories

Definition

Let $u : A \to B$ be a functor (Covariant functor, identity functor, composite functor, and contravariant functor). It is final when, for every $b \in B$ , the comma category $(b ↓ u)$ (Comma category, slice category, and coslice category) is nonempty and connected in the finite-zigzag sense of Isomorphism, groupoid, and connected category. Its objects are pairs $(a, β : b \to u (a))$ .

The functor $u$ is initial when $u^{o p} : A^{o p} \to B^{o p}$ is final (Opposite category $C^{o p}$ ). Equivalently, every $(u ↓ b)$ is nonempty and connected. Some sources call a final functor cofinal.

Depends on

Used by

Assuming Choice, precomposition with a final functor does not change colimits, and precomposition with an initial functor does not change limits Theorem

Dependency tree · two levels

5 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

T. Leinster, Basic Category Theory, Definition 6.3.1 (standard reference, not scraped)