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.

Sequential prespectra, spectra, and adjoint structure maps

Definition

Depends on

Dependency tree · 2 levels

Sources