DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Mapping cylinder and mapping cone

Definition

For a continuous $f : X \to Y$ , the unreduced mapping cylinder is $M_{f} = (Y ⨿ (X \times I)) / ((x, 0) \sim f (x)) .$ Write $j : X \to M_{f}$ , $j (x) = [x, 1]$ , and $r : M_{f} \to Y$ , $r (y) = y$ , $r ([x, s]) = f (x)$ . The formula for r descends continuously by For a quotient map $q : X \to Y$ , a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$ ; and a composite of quotient maps is a quotient map. The unreduced mapping cone is $C_{f} = M_{f} / j (X)$ , collapsing the entire free end to a point when it is nonempty. If X is empty this quotient is Y, with no extra cone point adjoined. These constructions do not collapse a basepoint track. Reduced cones and suspensions are defined separately below. Deformations always use the homotopy convention in Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints.

Depends on

Used by

Reduced cone suspension and cofiber sequence Definition
Mapping cone of a degree d circle map Example
Mapping cylinder factorization Theorem

Dependency tree · two levels

12 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

May, A Concise Course in Algebraic Topology (standard reference, not scraped)