LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Cone triangles satisfy TR3

Statement

The distinguished cone triangles in $K (A)$ satisfy TR3.

Facts & Assumptions

Given: A commuting square of first maps between two standard cone triangles.

Proof

1.1

Choose chain-map representatives $a : C \to C^{'}$ and $b : D \to D^{'}$ for the two vertical arrows. Commutativity in $K (A)$ means that there is a chain homotopy $H$ with $b f - f^{'} a = d H + H d .$ With the cone convention of Distinguished cone triangle in the homotopy category, the matrix $(\begin{array}{cc} b & H \\ 0 & a [1] \end{array}) : Cone (f) ⟶ Cone (f^{'})$ is a chain map.

given

2.1

Its composites with the cone inclusion and projection agree in $K (A)$ with the prescribed vertical arrows (the possible $H$ -terms are precisely homotopies). Thus its homotopy class is the required third component of a morphism of triangles; isomorphic replacements of standard cone triangles preserve the conclusion.

step 1.1given∎

Depends on

Used by

The homotopy category of an abelian category is triangulated Theorem

Dependency tree · two levels

5 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, Lemma 13.9.2 (standard reference, not scraped)