ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Two universal delta functors and their unique isomorphism

Example

Assume the Axiom of Dependent Choice.

Let $A$ and $B$ be abelian categories, suppose $A$ has enough projectives, and let $F : A \to B$ be additive and right exact. If $P$ and $Q$ are two supplied projective resolution data on all objects of $A$ , then the two universal homological delta functors $(L_{n}^{P} F)_{n \geq 0} and (L_{n}^{Q} F)_{n \geq 0}$ are uniquely isomorphic. After choosing the standard natural identifications $L_{0}^{P} F ≅ F ≅ L_{0}^{Q} F$ , the isomorphism is the unique one whose degree-zero component corresponds to ${id}_{F}$ .

Facts & Assumptions

Given: Two supplied projective resolution data $P$ and $Q$ on all objects of $A$ for the same right exact functor $F$ .

[L1]

Each of the two derived constructions is a universal homological delta functor (Derived functors are universal delta functors).

[L2]

Two universal delta functors with the same degree-zero term are uniquely isomorphic (Universal delta functors extending the same degree-zero functor are uniquely isomorphic).

[L3]

For every supplied projective resolution datum, the zeroth left derived functor is naturally isomorphic to the original right exact functor (Left derived functors form a homological delta functor).

Verification

technique · direct

1.1

By [L1], both $(L_{n}^{P} F)$ and $(L_{n}^{Q} F)$ are universal homological delta functors, and [L3] supplies their natural degree-zero identifications with $F$ .

L1L3given

2.1

Choose the natural degree-zero identifications with $F$ from step 1.1. Applying [L2] yields the unique isomorphism of delta functors whose degree-zero part corresponds under those identifications to ${id}_{F}$ ; its higher components are then forced.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)