False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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FALSE: mapping cone is a functor on the homotopy category with no extra data

Statement

Mapping cone defines a functor on the homotopy category with no extra choices.

Facts & Assumptions

Given: The zero map $0 : C_{∙} \to D_{∙}$ , where $C_{∙}$ is the stalk complex $Z [0]$ and $D_{∙}$ is the stalk complex $Z [1]$ .

[A1]

The statement refuted is: mapping cone defines a functor on the homotopy category with no extra choices.

[L1]

Chain homotopies of maps can alter the induced upper-triangular cone map (A chain homotopy).

[L2]

Strict functoriality is proved only on the arrow category of chain maps (Mapping cone is functorial on the arrow category of complexes).

Refutation

technique · direct

1.1

The identity square on the zero map admits two homotopies between the two zero composites: the zero homotopy and the degree-one map $t_{0} = 1_{Z}$ . They induce two cone endomorphisms of $Cone (0) = Z [1] \oplus Z [1],$ namely the identity matrix and $(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}) .$

A1L1givenalgebra

2.1

This cone complex has zero differential, so two endomorphisms are homotopic only when they are equal. The two matrices from step 1.1 are distinct, so a homotopy-category morphism does not determine a unique cone morphism without extra data. Therefore [A1] is false, and [L2] records the correct strict functoriality level.

A1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 13.9: Cones and termwise split sequences (standard reference, not scraped)