ExampleConstruction: Literature-sourcedVerification: AI-generatedaudited 2026-09-04

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.

A contractible simplicial cone

Example

Given a simplicial complex $K$ and a new vertex $a$ , its cone $a * K$ is obtained by adjoining to every simplex $σ$ the simplex ${a} \cup σ$ . The same formula used for a single simplex defines a degree- $1$ operator by setting $h [σ] = [a, σ]$ when $a$ is not already a vertex of $σ$ and $h [σ] = 0$ otherwise. The same cancellation check gives $\partial h + h \partial = id$ on the augmented simplicial chain complex.

Thus the augmented chain complex of a simplicial cone is contractible, so its reduced simplicial homology vanishes. In particular, a simplicial cone is a concrete combinatorial model of a contractible space.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Vidit Nanda, Computational Algebraic Topology, Lecture 02: Homotopy (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 03: Homology (standard reference, not scraped)