ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 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.

Extending a degree-zero natural transformation

Example

Let $S = (S_{n}, \partial^{S})$ be a homological delta functor, let $T = (T_{n}, \partial^{T})$ be an effaceable homological delta functor, and let $u_{0} : S_{0} \Rightarrow T_{0}$ be a natural transformation. For an object $A$ , choose an effacement $0 \to K \to P \overset{p}{\to} A \to 0$ with $T_{1} (p) = 0$ . Then the degree-one component of the universal extension is the unique map $u_{1} (A) : S_{1} (A) \to T_{1} (A)$ satisfying $\partial_{1}^{T} u_{1} (A) = u_{0} (K) \partial_{1}^{S},$ and this map is independent of the chosen effacement and compatible with the connecting morphisms.

Facts & Assumptions

Given: A degree-zero natural transformation $u_{0} : S_{0} \Rightarrow T_{0}$ and a chosen effacement of one object $A$ .

[L1]

One dimension shift defines the next-degree component from the chosen effacement (A partial morphism of delta functors extends through one dimension shift).

[L2]

The resulting component is independent of the effacing morphism and commutes with connecting maps (The effacement extension is independent of the effacing morphism, The effacement extension commutes with connecting morphisms).

[L3]

Derived-functor universality is built from exactly this extension mechanism (Derived functors are universal delta functors).

Verification

technique · direct

1.1L1given

The defining equation for $u_{1} (A)$ is exactly the homological case of [L1] with $n = 1$ .

2.1L2L3step 1.1∎

Item [L2] removes dependence on the chosen effacement and supplies the required compatibility with connecting morphisms, so the map from step 1.1 is the correct first higher component of the universal extension. This is the degree-one pattern used abstractly in [L3].

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)