False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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FALSE: a degree-zero natural transformation between delta functors always extends uniquely

Statement

False. Every degree-zero natural transformation between delta functors extends uniquely to a morphism of delta functors.

Facts & Assumptions

Given: The exact identity functor $E = {id}_{A b}$ .

[L1]

Extension from degree zero is the extra property called universality (Universal delta functor).

[L2]

A homological delta functor consists of additive functors, exact long sequences, and natural connecting maps (Homological delta functor).

Refutation

technique · direct

1.1

Define a homological delta functor $T$ on $A b$ by $T_{1} = E$ and $T_{n} = 0$ for $n \neq 1$ , with every connecting map zero. For each short exact sequence, the only nonzero part of its long sequence is $0 \to E (A) \to E (B) \to E (C) \to 0,$ which is exact because $E$ is exact; naturality is immediate. Thus [L2] applies.

L2givenconstruct

2.1

The unique degree-zero transformation $T_{0} = 0 \Rightarrow T_{0} = 0$ has at least two extensions $T \to T$ : the zero morphism and the identity morphism. They differ in degree $1$ , since $T_{1} = E \neq 0$ , but have the same degree-zero component. Therefore arbitrary delta functors do not have the unique-extension property [L1].

L1step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)