Alphabeta MathHow 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.
Subcomplexes, closures, stars, and links in a simplicial complex
Definition
Let be an abstract simplicial complex.
A subcomplex of is a subfamily that is itself closed under taking subsets.
If , its closure is the subcomplex More generally, if , its closure is
For a simplex , the closed star of in is and the open star is the union, after realization, of the relative interiors of simplices with .
The link of in is
Depends on
Used by
- Finite large spherical complexes, their almost-negative matrices, the metric flag condition, and links Definition
- Relative smooth triangulations, strong approximation and bundle fitting Definition
- Spherical Gram simplices and angular links of Euclidean faces Definition
- Residues, the compact chamber quotient, and the finite Coxeter sphere versus the contractible Davis cell Example
- Face coherence, global hat coordinates and a uniform star radius Lemma
- Face links of large metric flag complexes, and the inductive local CAT(1) criterion Lemma
- Finite simplicial constructible sheaves have an actual derived incidence model Lemma
- Finite simplicial open stars have explicit conical product charts Lemma
Dependency tree · one level
1 result within one dependency step 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 (standard reference, not scraped)
- Vidit Nanda, Computational Algebraic Topology, Lecture 01: Complexes (standard reference, not scraped)