PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Singular chains and singular homology are covariantly functorial

Statement

For each abelian group $G$ , the assignments $X \mapsto C_{∙} (X; G), f \mapsto f_{#}$ and $X \mapsto H_{n}^{s i n g} (X; G), f \mapsto H_{n} (f_{#})$ define covariant functors from topological spaces to chain complexes and to abelian groups, respectively. Equivalently, ${id}_{X, #} = {id}_{C_{∙} (X; G)}, (g \circ f)_{#} = g_{#} \circ f_{#},$ and for every $n \geq 0$ , $H_{n} ((g \circ f)_{#}) = H_{n} (g_{#}) \circ H_{n} (f_{#}), H_{n} ({id}_{X, #}) = {id}_{H_{n} (X; G)} .$

Facts & Assumptions

Given: An abelian group $G$ and continuous maps $f : X \to Y$ and $g : Y \to Z$ .

[L1]

The induced singular chain map sends a singular simplex $σ$ to $f \circ σ$ (The induced singular chain map of a continuous map).

[L2]

Induced singular chain maps commute with the singular boundaries (Induced singular chain maps commute with boundaries).

[L3]

Homology sends identity chain maps to identity maps and composite chain maps to composite homology maps (Homology respects identities and composition).

Proof

technique · direct

1.1L1given

For every singular simplex $σ$ in $X$ , [L1] gives ${id}_{X, #} (σ) = {id}_{X} \circ σ = σ$ and $(g \circ f)_{#} (σ) = (g \circ f) \circ σ = g \circ (f \circ σ) = g_{#} (f_{#} (σ))$ . By linearity, ${id}_{X, #}$ is the identity chain map and $(g \circ f)_{#} = g_{#} \circ f_{#}$ on singular chains.

2.1L2L3step 1.1∎

By [L2], every induced map $f_{#}$ is a chain map. Therefore step 1.1 gives identity and composition laws in the category of chain complexes, and [L3] transfers those same laws to singular homology in each degree.

Depends on

Used by

Dependency tree · two levels

12 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)