DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Barycentric subdivision operator

Definition

Define $S$ on a singular $0$ -simplex to be the identity. Recursively, for an $n$ -simplex $σ$ with $n > 0$ , set $S (σ) = b_{σ} S (\partial σ)$ , and extend $G$ -linearly to finite chains. It is the barycentric subdivision operator with the cone orientation of Barycenter and affine cone.

Depends on

Used by

Relative cup product for an excisive triad Definition
Subdivision prism homotopy Definition
A finite chain needing different subdivision depths on its simplices Example
First barycentric subdivision of a triangle Example
One fixed number of barycentric subdivisions makes every singular simplex cover small False statement
Mesh tends to zero under iterated subdivision Lemma
Barycentric subdivision and prism preserve smooth singular chains Proposition
Barycentric subdivision is a chain map Theorem
The cover-small inclusion is a chain homotopy equivalence Theorem

Dependency tree · two levels

3 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

Allen Hatcher, Algebraic Topology, Proposition 2.21 (standard reference, not scraped)