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.

Sequential prespectra, spectra, and adjoint structure maps

Definition

A sequential prespectrum $E$ consists of well-pointed based CGWH spaces $E_{n}$ , indexed by $n \geq 0$ , and based structure maps

$σ_{n} : S^{1} \land E_{n} ⟶ E_{n + 1} .$

The sphere coordinate is always written first. Its adjoint structure map is

${\tilde{σ}}_{n} : E_{n} ⟶ Ω E_{n + 1}, {\tilde{σ}}_{n} (x) (t) = σ_{n} (t \land x) .$

The loop space carries the compactly generated compact-open topology. The loop--suspension adjunction identifies $σ_{n}$ and ${\tilde{σ}}_{n}$ .

On this page a spectrum means an Omega-prespectrum: every ${\tilde{σ}}_{n}$ is a weak homotopy equivalence. This is terminology in the strict sequential setting only; no stable model structure or replacement functor is part of the definition.

Depends on

Used by

Pairings and unital multiplication of sequential prespectra Definition
Shift and suspension of sequential prespectra Definition
Stable homotopy groups of a sequential prespectrum Definition
Strict maps and structure-compatible homotopies of sequential prespectra Definition
Suspension and sphere prespectra Definition

Dependency tree · two levels

7 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)