DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Subdivision prism homotopy

Definition

Let $ι_{n} : Δ^{n} \to Δ^{n}$ be the identity affine simplex. Put $P_{0} = 0$ . Inductively, after $P_{m}$ has been defined for $m < n$ , set $P (λ) = λ_{#} P_{m}$ for every affine $m$ -simplex $λ : Δ^{m} \to Δ^{n}$ . Then define the linear $(n + 1)$ -chain $P_{n} = b_{n} (ι_{n} - P (\partial ι_{n})),$ where $b_{n} (-)$ is affine coning inside the convex simplex $Δ^{n}$ to its barycenter. For a singular simplex $σ : Δ^{n} \to X$ , set $T (σ) = σ_{#} P_{n}$ and extend $G$ -linearly. This degree-one operator is the subdivision prism homotopy.

Depends on

Dependency tree · 2 levels

Sources

Allen Hatcher, Algebraic Topology, Proposition 2.21