PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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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Strict prespectrum maps act functorially on stable homotopy groups

Statement

A strict map $f : E \to F$ of sequential prespectra induces, for every $k \in Z$ , a homomorphism

$π_{k} (f) : π_{k} (E) ⟶ π_{k} (F) .$

These maps preserve identities and composition. Structure-compatible homotopic strict maps induce the same homomorphism.

In this strict sequential-prespectrum setting, a stable weak equivalence is, by definition, a strict map $f$ for which $π_{k} (f)$ is an isomorphism for every $k \in Z$ .

Facts & Assumptions

[F1]

Strictness says $f_{n + 1} σ_{n} = τ_{n} (1_{S^{1}} \land f_{n})$ (Strict maps and structure-compatible homotopies of sequential prespectra).

[F2]

Based homotopic maps induce the same homomorphism on every higher homotopy group (Higher homotopy groups are functorial and based homotopy invariant).

[F3]

Finite-tail deletion gives the canonical colimit isomorphism proved in the preceding lemma (Stable homotopy colimits are independent of a cofinal tail).

Proof

Given: A strict map $f : E \to F$ and an integer $k$ as in the statement.

1.1

Construct the map of directed systems. Functoriality of higher homotopy gives $(f_{n})_{*} : π_{n + k} (E_{n}) \to π_{n + k} (F_{n})$ . Suspending a representative and using [F1] shows [F1, F2] $(f_{n + 1})_{*} b_{n}^{E} = b_{n}^{F} (f_{n})_{*} .$

Thus the level maps form a natural transformation of the two sequential systems. [F1, F2]

2.1

Descend to the colimit. Set $π_{k} (f) [n, x] = [n, (f_{n})_{*} x]$ . The equality in step 1.1 shows that equivalent representatives have equivalent images. Common-stage addition shows this map is a homomorphism. Different legal initial cutoffs give the same map under [F3].

F1F3step 1.1

3.1

Check functoriality and homotopy invariance. Identity and composite formulas hold at every level and hence in the colimit. If $H : f ≃ g$ is structure-compatible, then [F2] gives $(f_{n})_{*} = (g_{n})_{*}$ at every stage; their colimit maps are therefore equal. The last paragraph of the statement introduces terminology only. It does not assert a model structure, a replacement theorem, or a criterion involving levelwise equivalences. $□$

F2step 2.1

Depends on

Used by

Stabilizing a map between spheres Example
Scope boundary for stable homotopy theory Remark

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

J. P. May, A Concise Course in Algebraic Topology (standard reference, not scraped)
Allen Hatcher, Spectral Sequences in Algebraic Topology, Chapter 2 (standard reference, not scraped)