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

Used by

Ordinary barycentric subdivision cannot fix a nonconstant simplicial edge Counterexample
Canonical barycentric realization map Definition
Open and closed stars in a subdivision Definition
Oriented simplicial subdivision operator Definition
Relative derived subdivision of a finite simplicial pair Definition
Barycentric subdivision of an edge and triangle Example
An augmented simplicial cone has an explicit chain contraction Lemma
Last vertex map is carried by original simplices Lemma

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

C. R. F. Maunder, Algebraic Topology (standard reference, not scraped)