DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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 simplicial map and its geometric realization

Definition

Let $(V, K)$ and $(W, L)$ be abstract simplicial complexes. A function $f : V \to W$ is a simplicial map if $f (σ) : = {f (v) : v \in σ}$ is a simplex of $L$ whenever $σ$ is a simplex of $K$ .

The geometric realization of $f$ is the map $∣ f ∣ : ∣ K ∣ \to ∣ L ∣$ defined by $∣ f ∣ (α) (w) : = \sum_{v \in f^{- 1} (w)} α (v) .$ Because $α$ has finite support, the sum is finite. The support of $∣ f ∣ (α)$ is contained in $f (supp (α))$ , so it is again a simplex of $L$ .

Depends on

Used by

A vertex map need not be simplicial Counterexample
Contiguous simplicial maps Definition
The induced graded homomorphism of a simplicial map Definition
The realization of a simplicial map is continuous and functorial Lemma

Dependency tree · two levels

5 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

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
Vidit Nanda, Computational Algebraic Topology, Lecture 01: Complexes (standard reference, not scraped)