DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Strict maps and structure-compatible homotopies of sequential prespectra

Definition

Let $E = (E_{n}, σ_{n})$ and $F = (F_{n}, τ_{n})$ be sequential prespectra. A strict map of prespectra $f : E \to F$ is a family of based maps $f_{n} : E_{n} \to F_{n}$ such that, for every $n \geq 0$ ,

$f_{n + 1} \circ σ_{n} = τ_{n} \circ (1_{S^{1}} \land f_{n}) : S^{1} \land E_{n} ⟶ F_{n + 1} .$

A structure-compatible homotopy $H : f ≃ g$ is a family of based homotopies $H_{n} : E_{n} \times I \to F_{n}$ satisfying

$H_{n, 0} = f_{n}, H_{n, 1} = g_{n},$

and for which every time slice $H_{n, t}$ is strict:

$H_{n + 1, t} \circ σ_{n} = τ_{n} \circ (1_{S^{1}} \land H_{n, t}) (t \in I) .$

Thus this page uses the strict level category and this specified notion of homotopy; it does not identify arbitrary zigzags or introduce stable maps.

Depends on

Sequential prespectra, spectra, and adjoint structure maps

Used by

Strict prespectrum maps act functorially on stable homotopy groups Proposition

Dependency tree · two levels

3 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, Spectral Sequences in Algebraic Topology, Chapter 2 (standard reference, not scraped)