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.

Morphism of homological delta functors

Definition

Let $S = (S_{n}, \partial^{S})$ and $T = (T_{n}, \partial^{T})$ be homological delta functors. A morphism of homological delta functors $u : S \to T$ is a family of natural transformations $u_{n} : S_{n} \Rightarrow T_{n}, n \geq 0,$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ and every $n > 0$ , the square $\begin{array}{ccc} S_{n} (C) & \overset{\partial_{n}^{S}}{\to} & S_{n - 1} (A) \\ ↓^{u_{n} (C)} & ↓^{u_{n - 1} (A)} \\ T_{n} (C) & \overset{\partial_{n}^{T}}{\to} & T_{n - 1} (A) \end{array}$ commutes.

Equivalently, the degreewise natural transformations assemble into a morphism of the long exact sequences attached to every short exact sequence.

Morphism of homological delta functors

Definition

Depends on

Dependency tree · 2 levels

Sources