Alphabeta Math
PropositionStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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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.

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Simplicial homology is functorial

Statement

Every simplicial map f:K→L induces homomorphisms f∗:Hnsimp(K)→Hnsimp(L), and these induced maps respect identities and composition.

Proof

Given: Simplicial maps f:K→L and g:L→M.

1.1

The previous lemma shows that each f# is a chain map, so simplicial homology may be applied degreewise to obtain homomorphisms f∗.

given
1.2

On every oriented simplex, the identity simplicial map induces the identity chain map, and (g∘f)#=g#∘f# by direct inspection of the defining formula. Therefore the induced homology maps satisfy (id⁡K)∗=id⁡ and (g∘f)∗=g∗∘f∗.

given
2.1

Thus simplicial homology defines a functor from simplicial complexes and simplicial maps to graded abelian groups.

step 1.1step 1.2∎

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