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.

Homological delta functor

Definition

Let $A$ and $B$ be abelian categories. A homological delta functor from $A$ to $B$ is a family of additive functors $T_{n} : A \to B, n \geq 0,$ together with, for every short exact sequence $0 \to A \to B \to C \to 0$ in the sense of Exact sequence and short exact sequence in an abelian category, a family of morphisms $\partial_{n} : T_{n} (C) \to T_{n - 1} (A), n > 0,$ such that:

the sequence $\dots \to T_{n} (A) \to T_{n} (B) \to T_{n} (C) \overset{\partial_{n}}{\to} T_{n - 1} (A) \to T_{n - 1} (B) \to T_{n - 1} (C) \to \dots \to T_{0} (C) \to 0$ is exact, and
for every morphism between short exact sequences, the connecting squares $\begin{array}{ccc} T_{n} (C) & \overset{\partial_{n}}{\to} & T_{n - 1} (A) \\ ↓ & ↓ \\ T_{n} (C^{'}) & \overset{\partial_{n}^{'}}{\to} & T_{n - 1} (A^{'}) \end{array}$ commute.

Thus the structure consists of both the long exact sequence and the naturality of its connecting maps.

Homological delta functor

Definition

Depends on

Dependency tree · 2 levels

Sources