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

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 of an abstract simplicial complex

Definition

The barycentric subdivision of $K$ is $sd K = Δ F (K)$ , using Face poset and order complex. Thus its vertices are the nonempty faces $σ$ of $K$ , and its nonempty simplices are strict chains $σ_{0} ⊊ \dots ⊊ σ_{q}$ .

A subcomplex $A \subseteq K$ gives $F (A) \subseteq F (K)$ with the induced order; hence every chain in $F (A)$ is a chain in $F (K)$ and $sd A$ is a subcomplex. The vertex associated to ${v}$ is distinct as a label from $v$ , though its geometric position will be the same. There is no vertex for the empty face; the complex with no vertices subdivides to itself.

Source locators

2.5.7–2.5.10, pp.49–52.

Depends on

Face poset and order complex

Dependency tree · 2 levels

Sources

C. R. F. Maunder, Algebraic Topology