LemmaStatement: AI-generatedProof: AI-generated

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.

Stable rotation and exact coherent products

PDF p. 3: Filtered S1 spectra: stable enhancement prerequisites

Statement

In $S p$ every small coherent diagram has a homotopy limit and colimit. For $f : A \to B$ define $fib (f) = A \times_{B} 0$ and $cofib (f) = B ⨿_{A} 0$ . Then $cofib (f) ≃ fib (f) [1]$ . A fiber sequence $F \to A \to B$ , including its nullhomotopy and defining pullback square, extends to the cofiber triangle $F \to A \to B \to F [1]$ . Products are coherent homotopy products. Smash with a fixed spectrum preserves finite limits and finite colimits in either variable.

Facts & Assumptions

Given: Spectra, a map $f$ and a small indexing infinity-category. Write $0$ for the zero object and $[1]$ for suspension, $E [1] = 0 ⨿_{E} 0$ .

Goal: Establish rotation, products and separate exactness of smash.

[F1]

The stable enhancement gives all small limits and colimits, equality of pushouts and pullbacks, and a symmetric monoidal smash preserving colimits separately. External stable enhancement and smash product of spectra

Proof

1.1F1

The limits and colimits exist by [F1] and are homotopy limits and colimits because their universal properties are equivalences of mapping spaces. In particular, the limit over a discrete small indexing category is the homotopy product, with the coherences supplied by the limit construction.

1.2F1given

The square defining $F$ as the fiber of $A \to B$ is a pullback and hence a pushout. Thus $B ≃ A ⨿_{F} 0$ . Pasting pushouts gives $B ⨿_{A} 0 ≃ (A ⨿_{F} 0) ⨿_{A} 0 ≃ 0 ⨿_{F} 0 = F [1]$ . The induced map $B \to F [1]$ completes the stated cofiber triangle. Taking $F = fib (f)$ gives $cofib (f) ≃ fib (f) [1]$ .

2.1F1step 1.1step 1.2∎

Smash with a fixed spectrum preserves colimits and therefore the zero object and pushout squares. A pullback square is a pushout square, so its image is again both a pushout and a pullback. Preservation of the zero object and pullbacks gives preservation of finite limits; finite colimits are preserved by assumption. The same argument applies in the other variable. This proves the statement.

Depends on

External stable enhancement and smash product of spectra

Used by · 1

Stable spectra and exact triangles

Dependency tree · 2 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.