ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

How statement and proof provenance work

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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.

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The mapping cylinder of an inclusion of two-term complexes

Example

Let $C_{∙} = Z [0]$ and let $D_{∙}$ be the two-term complex $0 \to Z \overset{1}{\to} Z \to 0,$ with the right copy in degree $0$ . The inclusion $f : C_{∙} \to D_{∙}$ into degree $0$ has mapping cylinder $Cyl (f)_{1} ≅ Z \oplus Z, Cyl (f)_{0} ≅ Z \oplus Z,$ with differentials read directly from the cylinder formula, and the projection $p : Cyl (f) \to D$ is a homotopy equivalence.

Facts & Assumptions

Given: The inclusion $f : Z [0] \to (0 \to Z \overset{1}{\to} Z \to 0)$ .

[L1]

The mapping-cylinder terms and differential are given explicitly by The mapping cylinder of a chain map.

[L2]

The mapping cylinder factors a chain map through a homotopy equivalence (The mapping cylinder factors a chain map).

Verification

technique · direct

1.1L1givenalgebra

Applying [L1] degreewise leaves the displayed two copies of $Z^{2}$ in degrees $1$ and $0$ ; all other terms vanish.

2.1L2step 1.1algebra∎

The same construction comes with maps $i$ and $p$ , and [L2] identifies $p$ as a chain-homotopy equivalence. So this example is an explicit two-term instance of the general factorization theorem.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)