Alphabeta MathHow 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.
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External singular realization comparison
PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites
Statement
For a based space X, the evaluation ε:|Sing X|→X is a natural weak homotopy equivalence, and realization is left adjoint to the singular complex in the pointed categories. The basepoint of Sing X is the constant singular simplex at the basepoint in each degree. A weak homotopy equivalence induces a bijection on path components and isomorphisms on all homotopy groups at every basepoint.
Source contract
Given: A based topological space X.
Goal: Supply the realization and singular-complex comparison used for sequential prespectra.
[F1] May, Chapter 16, §§2–4, gives the evaluation theorem and realization–singular adjunction at the cited locator.
External input
Import [F1], including its pointed specialization. This is an external theorem; no local reconstruction of the comparison is claimed.
Depends on
Dependency tree · 1 level
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.
External dependencies
The evaluation from the geometric realization of the singular complex of a space to that space is a natural weak homotopy equivalence; realization and singular complex are adjoint.
Hypotheses: A topological space; for the pointed version, a chosen basepoint and based maps.
Reading limits: The current author inspected the construction and theorem statement in §2 and the beginning of the proof in §3. The supplied authoring assistance reports reading §§2–4 and notes that May sketches the weak-equivalence proof. The pointed adjunction and remaining foundational details are imported, not reconstructed here.
Sources
- J. P. May, A Concise Course in Algebraic Topology · Chapter 16, §§2–4, printed pp. 124–127, especially the second theorem of §2 and the realization–singular adjunction in §4