TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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.

The mapping cylinder factors a chain map

Statement

For every chain map $f : C_{∙} \to D_{∙}$ , the mapping cylinder gives a factorization $C_{∙} \overset{i}{\to} Cyl (f)_{∙} \overset{p}{\to} D_{∙}$ with $p i = f$ , where $i$ is degreewise split monic and $p$ is a chain-homotopy equivalence.

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ .

[L1]

The maps for the mapping cylinder are $i (x) = (x, 0, 0), p (x, y, z) = f (x) + y, j (y) = (0, y, 0)$ (The mapping cylinder of a chain map).

[L2]

A chain homotopy equivalence is a chain map admitting a homotopy inverse (A chain homotopy equivalence).

[L3]

Identities and composites of chain maps are chain maps (Identities and composites of chain maps are chain maps).

Proof

technique · direct

1.1

By [L1], $p i = f$ and $p j = 1_{D}$ . The map $r_{n} : Cyl (f)_{n} \to C_{n}$ , $r_{n} (x, y, z) = x$ , satisfies $r_{n} i_{n} = 1_{C_{n}}$ , so $i_{n}$ is split monic in every degree.

L1givenalgebra

2.1

Define $H_{n} : Cyl (f)_{n} \to Cyl (f)_{n + 1}$ by $H_{n} (x, y, z) : = (0, 0, x) .$ Using the cylinder differential, one computes $d H + H d = 1_{Cyl (f)} - j p .$ Thus $j$ is a homotopy inverse for $p$ , so [L2] shows that $p$ is a chain-homotopy equivalence; [L3] guarantees all composites involved are chain maps.

L2L3step 1.1constructalgebra∎

Depends on

Used by

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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)