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.

Universal delta functor

Definition

Let $T$ be a delta functor.

If $T = (T_{n}, \partial^{T})$ is homological, then $T$ is universal when for every homological delta functor $S = (S_{n}, \partial^{S})$ and every natural transformation $u_{0} : S_{0} \Rightarrow T_{0},$ there exists a unique morphism of homological delta functors $u : S \to T$ whose degree-zero component is $u_{0}$ .

If $T = (T^{n}, \partial_{T})$ is cohomological, then $T$ is universal when for every cohomological delta functor $S = (S^{n}, \partial_{S})$ and every natural transformation $u^{0} : T^{0} \Rightarrow S^{0},$ there exists a unique morphism of cohomological delta functors $u : T \to S$ whose degree-zero component is $u^{0}$ .

So universality says that the higher-degree components are forced by the degree-zero data and the delta-functor axioms.

Universal delta functor

Definition

Depends on

Dependency tree · 2 levels

Sources