DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Left total derived functor on the bounded above derived category

Definition

Let $A$ and $B$ be abelian categories and let $F : A \to B$ be additive. Supply bounded-above projective replacements $p_{X} : P_{X} \to X$ and the hypotheses for Projective complexes model the bounded above derived category. Write $T = Q_{B} K (F)$ . The left total derived functor is $L F : D^{-} (A) \to D^{-} (B)$ with $L F (X) = Q_{B} F (P_{X})$ , its maps obtained from the unique homotopy classes between projective models. Its augmentation is $ϵ : L F Q_{A} \to T$ , induced by $F (p_{X})$ .

The defining universal property is terminal: for every functor $G : D^{-} (A) \to D^{-} (B)$ and natural $γ : G Q_{A} \to T$ , there is a unique natural $μ : G \to L F$ with $ϵ \circ (μ Q_{A}) = γ$ . Additive has the meaning of Additive functor.

Depends on

Used by

Dependency tree · two levels

8 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

10.5.1–10.5.8, pp. 391–393; restrict to supplied replacement data (standard reference, not scraped)